Principles of minimum variance robust adaptive beamforming design

Robustness is typically understood as an ability of adaptive beamforming algorithm to achieve high performance in the situations with imperfect, incomplete, or erroneous knowledge about the source, propagation media, and antenna array. It is also desired to achieve high performance with as little as possible prior information. In the last decade, several fruitful principles to minimum variance distortionless response (MVDR) robust adaptive beamforming (RAB) design have been developed and successfully applied to solve a number of problems in a wide range of applications. Such principles of MVDR RAB design are summarized here in a single paper. Prof. Gershman has actively participated in the development and applications of a number of such MVDR RAB design principles

Adaptive matched field processing in an uncertain propagation environment

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution January 1992Adaptive array processing algorithms have achieved widespread use because they are very effective at rejecting unwanted signals (i.e., controlling sidelobe levels) and in general have very good resolution (i.e., have narrow mainlobes). However, many adaptive high-resolution array processing algorithms suffer a significant degradation in performance in the presence of environmental mismatch. This sensitivity to environmental mismatch is of particular concern in problems such as long-range acoustic array processing in the ocean where the array processor's knowledge of the propagation characteristics of the ocean is imperfect. An Adaptive Minmax Matched Field Processor has been developed which combines adaptive matched field processing and minmax approximation techniques to achieve the effective interference rejection characteristic of adaptive processors while limiting the sensitivity of the processor to environmental mismatch. The derivation of the algorithm is carried out within the framework of minmax signal processing. The optimal array weights are those which minimize the maximum conditional mean squared estimation error at the output of a linear weight-and-sum beamformer. The error is conditioned on the propagation characteristics of the environment and the maximum is evaluated over the range of environmental conditions in which the processor is expected to operate. The theorems developed using this framework characterize the solutions to the minmax array weight problem, and relate the optimal minmax array weights to the solution to a particular type of Wiener filtering problem. This relationship makes possible the development of an efficient algorithm for calculating the optimal minmax array weights and the associated estimate of the signal power emitted by a source at the array focal point. An important feature of this algorithm is that it is guarenteed to converge to an exact solution for the array weights and estimated signal power in a finite number of iterations. The Adaptive Minmax Matched Field Processor can also be interpreted as a two-stage Minimum Variance Distortionless Response (MVDR) Matched Field Processor. The first stage of this processor generates an estimate of the replica vector of the signal emitted by a source at the array focal point, and the second stage is a traditional MVDR Matched Field Processor implemented using the estimate of the signal replica vector. Computer simulations using several environmental models and types of environmental uncertainty have shown that the resolution and interference rejection capability of the Adaptive Minmax Matched Field Processor is close to that of a traditional MVDR Matched Field Processor which has perfect knowledge of the characteristics of the propagation environment and far exceeds that of the Bartlett Matched Field Processor. In addition, the simulations show that the Adaptive Minmax Matched Field Processor is able to maintain it's accuracy, resolution and interference rejection capability when it's knowledge of the environment is only approximate, and is therefore much less sensitive to environmental mismatch than is the traditional MVDR Matched Field Processor.The National Science Foundation, the General Electric Foundation, the Office of Naval Research, the Defense Advanced Research Projects Agency, and the Woods Hole Oceanographic Institution

Applied stochastic eigen-analysis

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2007The first part of the dissertation investigates the application of the theory of large random matrices to high-dimensional inference problems when the samples are drawn from a multivariate normal distribution. A longstanding problem in sensor array processing is addressed by designing an estimator for the number of signals in white noise that dramatically outperforms that proposed by Wax and Kailath. This methodology is extended to develop new parametric techniques for testing and estimation. Unlike techniques found in the literature, these exhibit robustness to high-dimensionality, sample size constraints and eigenvector misspecification. By interpreting the eigenvalues of the sample covariance matrix as an interacting particle system, the existence of a phase transition phenomenon in the largest (“signal”) eigenvalue is derived using heuristic arguments. This exposes a fundamental limit on the identifiability of low-level signals due to sample size constraints when using the sample eigenvalues alone. The analysis is extended to address a problem in sensor array processing, posed by Baggeroer and Cox, on the distribution of the outputs of the Capon-MVDR beamformer when the sample covariance matrix is diagonally loaded. The second part of the dissertation investigates the limiting distribution of the eigenvalues and eigenvectors of a broader class of random matrices. A powerful method is proposed that expands the reach of the theory beyond the special cases of matrices with Gaussian entries; this simultaneously establishes a framework for computational (non-commutative) “free probability” theory. The class of “algebraic” random matrices is defined and the generators of this class are specified. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue distribution and, for a subclass, the limiting conditional “eigenvector distribution.” The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. The method is applied to predict the deterioration in the quality of the sample eigenvectors of large algebraic empirical covariance matrices due to sample size constraints.I am grateful to the National Science Foundation for supporting this work via grant DMS-0411962 and the Office of Naval Research Graduate Traineeship awar

MVDR broadband beamforming using polynomial matrix techniques

This thesis addresses the formulation of and solution to broadband minimum variance distortionless response (MVDR) beamforming. Two approaches to this problem are considered, namely, generalised sidelobe canceller (GSC) and Capon beamformers. These are examined based on a novel technique which relies on polynomial matrix formulations. The new scheme is based on the second order statistics of the array sensor measurements in order to estimate a space-time covariance matrix. The beamforming problem can be formulated based on this space-time covariance matrix. Akin to the narrowband problem, where an optimum solution can be derived from the eigenvalue decomposition (EVD) of a constant covariance matrix, this utility is here extended to the broadband case. The decoupling of the space-time covariance matrix in this case is provided by means of a polynomial matrix EVD. The proposed approach is initially exploited to design a GSC beamformer for a uniform linear array, and then extended to the constrained MVDR, or Capon, beamformer and also the GSC with an arbitrary array structure. The uniqueness of the designed GSC comes from utilising the polynomial matrix technique, and its ability to steer the array beam towards an off-broadside direction without the pre-steering stage that is associated with conventional approaches to broadband beamformers. To solve the broadband beamforming problem, this thesis addresses a number of additional tools. A first one is the accurate construction of both the steering vectors based on fractional delay filters, which are required for the broadband constraint formulation of a beamformer, as for the construction of the quiescent beamformer. In the GSC case, we also discuss how a block matrix can be obtained, and introduce a novel paraunitary matrix completion algorithm. For the Capon beamformer, the polynomial extension requires the inversion of a polynomial matrix, for which a residue-based method is proposed that offers better accuracy compared to previously utilised approaches. These proposed polynomial matrix techniques are evaluated in a number of simulations. The results show that the polynomial broadband beamformer (PBBF) steersthe main beam towards the direction of the signal of interest (SoI) and protects the signal over the specified bandwidth, and at the same time suppresses unwanted signals by placing nulls in their directions. In addition to that, the PBBF is compared to the standard time domain broadband beamformer in terms of their mean square error performance, beam-pattern, and computation complexity. This comparison shows that the PBBF can offer a significant reduction in computation complexity compared to its standard counterpart. Overall, the main benefits of this approach include beam steering towards an arbitrary look direction with no need for pre-steering step, and a potentially significant reduction in computational complexity due to the decoupling of dependencies of the quiescent beamformer, blocking matrix, and the adaptive filter compared to a standard broadband beamformer implementation.This thesis addresses the formulation of and solution to broadband minimum variance distortionless response (MVDR) beamforming. Two approaches to this problem are considered, namely, generalised sidelobe canceller (GSC) and Capon beamformers. These are examined based on a novel technique which relies on polynomial matrix formulations. The new scheme is based on the second order statistics of the array sensor measurements in order to estimate a space-time covariance matrix. The beamforming problem can be formulated based on this space-time covariance matrix. Akin to the narrowband problem, where an optimum solution can be derived from the eigenvalue decomposition (EVD) of a constant covariance matrix, this utility is here extended to the broadband case. The decoupling of the space-time covariance matrix in this case is provided by means of a polynomial matrix EVD. The proposed approach is initially exploited to design a GSC beamformer for a uniform linear array, and then extended to the constrained MVDR, or Capon, beamformer and also the GSC with an arbitrary array structure. The uniqueness of the designed GSC comes from utilising the polynomial matrix technique, and its ability to steer the array beam towards an off-broadside direction without the pre-steering stage that is associated with conventional approaches to broadband beamformers. To solve the broadband beamforming problem, this thesis addresses a number of additional tools. A first one is the accurate construction of both the steering vectors based on fractional delay filters, which are required for the broadband constraint formulation of a beamformer, as for the construction of the quiescent beamformer. In the GSC case, we also discuss how a block matrix can be obtained, and introduce a novel paraunitary matrix completion algorithm. For the Capon beamformer, the polynomial extension requires the inversion of a polynomial matrix, for which a residue-based method is proposed that offers better accuracy compared to previously utilised approaches. These proposed polynomial matrix techniques are evaluated in a number of simulations. The results show that the polynomial broadband beamformer (PBBF) steersthe main beam towards the direction of the signal of interest (SoI) and protects the signal over the specified bandwidth, and at the same time suppresses unwanted signals by placing nulls in their directions. In addition to that, the PBBF is compared to the standard time domain broadband beamformer in terms of their mean square error performance, beam-pattern, and computation complexity. This comparison shows that the PBBF can offer a significant reduction in computation complexity compared to its standard counterpart. Overall, the main benefits of this approach include beam steering towards an arbitrary look direction with no need for pre-steering step, and a potentially significant reduction in computational complexity due to the decoupling of dependencies of the quiescent beamformer, blocking matrix, and the adaptive filter compared to a standard broadband beamformer implementation

Efficient Robust Adaptive Beamforming Algorithms for Sensor Arrays

Sensor array processing techniques have been an important research area in recent years. By using a sensor array of a certain configuration, we can improve the parameter estimation accuracy from the observation data in the presence of interference and noise. In this thesis, we focus on sensor array processing techniques that use antenna arrays for beamforming, which is the key task in wireless communications, radar and sonar systems. Firstly, we propose a low-complexity robust adaptive beamforming (RAB) technique which estimates the steering vector using a Low-Complexity Shrinkage-Based Mismatch Estimation (LOCSME) algorithm. The proposed LOCSME algorithm estimates the covariance matrix of the input data and the interference-plus-noise covariance (INC) matrix by using the Oracle Approximating Shrinkage (OAS) method. Secondly, we present cost-effective low-rank techniques for designing robust adaptive beamforming (RAB) algorithms. The proposed algorithms are based on the exploitation of the cross-correlation between the array observation data and the output of the beamformer. Thirdly, we propose distributed beamforming techniques that are based on wireless relay systems. Algorithms that combine relay selections and SINR maximization or Minimum Mean-Square- Error (MMSE) consensus are developed, assuming the relay systems are under total relay transmit power constraint. Lastly, we look into the research area of robust distributed beamforming (RDB) and develop a novel RDB approach based on the exploitation of the cross-correlation between the received data at the relays and the destination and a subspace projection method to estimate the channel errors, namely, the cross-correlation and subspace projection (CCSP) RDB technique, which efficiently maximizes the output SINR and minimizes the channel errors. Simulation results show that the proposed techniques outperform existing techniques in various performance metrics

Adaptive matched field processing in an uncertain propagation environment

Also issued as Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineeing and Woods Hole Oceanographic Institute, 1991.Includes bibliographical references (p. 171-173).Supported by DARPA. N00014-89-J-1489 Supported by NSF. MIP 87-14969 Supported by a General Electric Foundation Graduate Fellowship in Electrical Engineering, and a National Science Foudnation Graduate Fellowship.James C. Preisig

Polynomial matrix algebra with applications

[Abstract unavailable

3-D Beamspace ML Based Bearing Estimator Incorporating Frequency Diversity and Interference Cancellation

The problem of low-angle radar tracking utilizing an array of antennas is considered. In the low-angle environment, echoes return from a low flying target via a specular path as well as a direct path. The problem is compounded by the fact that the two signals arrive within a beamwidth of each other and are usually fully correlated, or coherent. In addition, the SNR at each antenna element is typically low and only a small number of data samples, or snapshots, is available for processing due to the rapid movement of the target. Theoretical studies indicates that the Maximum Likelihood (ML) method is the only reliable estimation procedure in this type of scenario. However, the classical ML estimator involves a multi-dimensional search over a multi-modal surface and is consequently computationally burdensome. In order to facilitate real time processing, we here propose the idea of beamspace domain processing in which the element space snapshot vectors are first operated on by a reduced Butler matrix composed of three orthogonal beamforming weight vectors facilitating a simple, closed-form Beamspace Domain ML (BDML) estimator for the direct and specular path angles. The computational simplicity of the method arises from the fact that the respective beams associated with the three columns of the reduced Butler matrix have all but three nulls in common. The performance of the BDML estimator is enhanced by incorporating the estimation of the complex reflection coefficient and the bisector angle, respectively, for the symmetric and nonsymmetric multipath cases. To minimize the probability of track breaking, the use of frequency diversity is incorporated. The concept of coherent signal subspace processing is invoked as a means for retaining the computational simplicity of single frequency operation. With proper selection of the auxiliary frequencies, it is shown that perfect focusing may be achieved without iterating. In order to combat the effects of strong interfering sources, a novel scheme is presented for adaptively forming the three beams which retains the feature of common nulls