A CLT on the SNR of Diagonally Loaded MVDR Filters

This paper studies the fluctuations of the signal-to-noise ratio (SNR) of minimum variance distorsionless response (MVDR) filters implementing diagonal loading in the estimation of the covariance matrix. Previous results in the signal processing literature are generalized and extended by considering both spatially as well as temporarily correlated samples. Specifically, a central limit theorem (CLT) is established for the fluctuations of the SNR of the diagonally loaded MVDR filter, under both supervised and unsupervised training settings in adaptive filtering applications. Our second-order analysis is based on the Nash-Poincar\'e inequality and the integration by parts formula for Gaussian functionals, as well as classical tools from statistical asymptotic theory. Numerical evaluations validating the accuracy of the CLT confirm the asymptotic Gaussianity of the fluctuations of the SNR of the MVDR filter.Comment: This is a corrected version of the paper that will appear at IEEE Transactions on Signal Processing September 201

Robust Recursive Steering Vector Estimation and Adaptive Beamforming under Sensor Uncertainties

published_or_final_versio

A directionally tunable but frequency-invariant beamformer on an acoustic velocity-sensor triad to enhance speech perception

Herein investigated are computationally simple microphone-array beamformers that are independent of the frequency-spectra of all signals, all interference, and all noises. These beamformers allow the listener to tune the desired azimuth-elevation “look direction.” No prior information is needed of the interference. These beamformers deploy a physically compact triad of three collocated but orthogonally oriented velocity sensors. These proposed schemes’ efficacy is verified by a jury test, using simulated data constructed with Mandarin Chinese (a.k.a. Putonghua) speech samples. For example, a desired speech signal, originally at a very adverse signal-to-interference-and-noise power ratio (SINR) of -30 dB, may be processed to become fully intelligible to the jury

Toeplitz Covariance Matrix Estimation for Adaptive Beamforming and Ultrasound Imaging

In recent years, adaptive beamformers have been researched more extensively, to be able to use it in the application of medical ultrasound imaging. The adaptive beamformers can provide a higher resolution and better contrasts in the resulting images than non-adaptive, and most commonly used, delay-and-sum beamformer. The difficulties of applying adaptive beamformers to ultrasound imaging can e.g. be numerical complexity, stability of statistics, coherent sources, or the robustness of the beamformer. Methods to handle these types of difficulties have been researched and successfully applied to utilize the performance advantages provided by adaptive beamformers. Why is it important to prevent these types of errors? The beamformer is used in the image formation stage, and errors that occur at this stage are difficult to get rid of even for the most sophisticated imaging software. We have in this thesis investigated methods that attempt to force the estimate of the covariance matrix to become a Toeplitz matrix. A Toeplitz matrix has equal elements along its diagonals, and has several useful properties that are desirable in array processing. Assuming the Toeplitz structure can be achieved, this is because of the spatial stationarity in the received data, it can be applied in medical ultrasound imaging. Three different methods are proposed to reach the desired Toeplitz structure in this thesis; IAAAPES, Adaptive Spatial Averaging and Spatial Convolution. The three methods for making the covariance matrix Toeplitz will be compared and analyzed to other known adaptive beamformers to detect their strengths and weaknesses

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

Performance analysis of the dominant mode rejection beamformer

In array signal processing over challenging environments, due to the non-stationarity nature of data, it is difficult to obtain enough number of data snapshots to construct an adaptive beamformer (ABF) for detecting weak signal embedded in strong interferences. One type of adaptive method targeting for such applications is the dominant mode rejection (DMR) method, which uses a reshaped eigen-decomposition of sample covariance matrix (SCM) to define a subspace containing the dominant interferers to be rejected, thereby allowing it to detect weak signal in the presence of strong interferences. The DMR weight vector takes a form similar to the adaptive minimum variance distortion-less response (MVDR), except with the SCM being replaced by the DMR-SCM. This dissertation studies the performance of DMR-ABF by deriving the probability density functions of three important metrics: notch depth (ND), white noise gain (WNG), and signal-to-interference-and-noise ratio (SINR). The analysis contains both single interference case and multiple interference case, using subspace transformation and the random matrix theory (RMT) method for deriving and verifying the analytical results. RMT results are used to approximate the random matrice. Finally, the analytical results are compared with RMT Monte-Carlo based empirical results

Array signal processing robust to pointing errors

The objective of this thesis is to design computationally efficient DOA (direction-of- arrival) estimation algorithms and beamformers robust to pointing errors, by harnessing the antenna geometrical information and received signals. Initially, two fast root-MUSIC-type DOA estimation algorithms are developed, which can be applied in arbitrary arrays. Instead of computing all roots, the first proposed iterative algorithm calculates the wanted roots only. The second IDFT-based method obtains the DOAs by scanning a few circles in parallel and thus the rooting is avoided. Both proposed algorithms, with less computational burden, have the asymptotically similar performance to the extended root-MUSIC. The second main contribution in this thesis is concerned with the matched direction beamformer (MDB), without using the interference subspace. The manifold vector of the desired signal is modeled as a vector lying in a known linear subspace, but the associated linear combination vector is otherwise unknown due to pointing errors. This vector can be found by computing the principal eigen-vector of a certain rank-one matrix. Then a MDB is constructed which is robust to both pointing errors and overestimation of the signal subspace dimension. Finally, an interference cancellation beamformer robust to pointing errors is considered. By means of vector space projections, much of the pointing error can be eliminated. A one-step power estimation is derived by using the theory of covariance fitting. Then an estimate-and-subtract interference canceller beamformer is proposed, in which the power inversion problem is avoided and the interferences can be cancelled completely

Temporal and Spatial Interference Mitigation Strategies to Improve Radar Data Quality

The microwave band is well suited to wireless applications, including radar, communications, and electronic warfare. While radar operations currently have priority in a portion of the microwave band, wireless companies are lobbying to change that; such a change would force current operators into a smaller total bandwidth. Interference would occur, and has already occurred at the former National Weather Radar Testbed Phased Array Radar. The research in this dissertation was motivated by this interference --- it occurred even without a change to radar's primacy in the microwave band. If microwave operations had to squeeze into a smaller overall bandwidth, such interference, whether originating from other radars or some other source, would only become more common. The radio frequency interference (RFI) present at the National Weather Radar Testbed Phased Array Radar altered the statistical properties at certain locations, causing targets to be erroneously detected. While harmless enough in clear air, it could affect National Weather Service decisions if it occurred during a weather event. The initial experiments, covered in Chapter 2, used data comprised of a single channel of in-phase and quadrature (IQ) data, reflecting the resources available to the National Weather Service's weather radar surveillance network. A new algorithm, the Interference Spike Detection Algorithm, was developed with these restrictions in mind. This new algorithm outperforms several interference detection algorithms developed by industry. Tests on this data examined algorithm performance quantitatively, using real and simulated weather data and radio frequency interference. Additionally, machine learning classification algorithms were employed for the first time to the RFI classification problem and it was found that, given enough resources, machine learning had the potential to perform even better than the other temporal algorithms. Subsequent experiments, covered in Chapter 3, used spatial data from phased arrays and looked at methods of interference mitigation that leveraged this spatial data. Specifically, adaptive beamforming techniques could be used to mitigate interference and improve data quality. A variety of adaptive digital beamforming techniques were evaluated in terms of their performance at interference mitigation for a communications task. Additionally, weather radar data contaminated with ground clutter was collected from the sidelobe canceller channels of the former National Weather Radar Testbed Phased Array Radar and, using the reasoning that ground clutter is simply interference from the ground, adaptive digital beamforming was successfully employed to mitigate the impact of ground clutter and restore the data to reflect the statistics of the underlying weather data. Tests on digital equalization, covered in Chapter 4, used data from a prototype receiver for Horus, a digital phased array radar under development at the University of Oklahoma. The data suffered from significant channel mismatch, which can severely negatively impact the performance of phased arrays. Equalization, implemented both via older digital filter design methods and, for the first time, via newer machine learning regression methods, was able to improve channel matching. When used before adaptive digital beamforming, it was found that digital equalization always improved system performance