Christ D. Richmond

Performance of a class of adaptive detection algorithms in nonhomogeneous environments

A two-dimensional (2-D) adaptive sidelobe blanker (ASB) detection algorithm was developed through experimentation as an extenuate for false alarms caused by undernulled interference encountered when applying the adaptive matched filter (AMF) in nonhomogeneous environments. The algorithms utility has been demonstrated empirically. Considering theoretic performance analyses of the ASB detection algorithm as well as the AMF generalized likelihood ratio test (GLRT), and the adaptive cosine estimator (ACE), under nonideal conditions, can become fairly intractable rather quickly, especially in an adaptive processing context involving covariance estimation. In this paper, however, we have developed and exploited a theoretic framework through which the performance of these algorithms under nonhomogeneous conditions can be examined theoretically. It is demonstrated through theoretic analysis that in the presence of undernulled interference, the ASB is a pliable false alarm regulatory (FAR) detector that maintains good target sensitivity. A viable method of ASB threshold selection is also presented and demonstrated.

Performance of the adaptive sidelobe blanker detection algorithm in homogeneous environments

The adaptive sidelobe blanker (ASB) algorithm is a two-stage detector consisting of a first stage adaptive matched filter (AMF) detector followed by a second-stage detector called the adaptive coherence (or cosine) estimator (ACE). Only those data test cells that survive both detection thresholdings are declared signal (target) bearing. We provide exact novel closed-form expressions for the resulting probability of detection (PD) and false alarm (PFA) for the ASB algorithm and demonstrate that under homogeneous data conditions with no signal array response mismatch that (i) the ASB is a constant false alarm rate (CFAR) algorithm, (ii) the ASB has a higher or commensurate PD for a given PFA than both the AMF and the ACE, and (iii) the ASB has an overall performance that is commensurate with Kellys (1986) benchmark generalized likelihood ratio test (GLRT). A compact statistical summary is derived providing distributions and dependencies among the GLRT, AMF, and the ACE decision statistics.

Capon algorithm mean-squared error threshold SNR prediction and probability of resolution

Below a specific threshold signal-to-noise ratio (SNR), the mean-squared error (MSE) performance of signal parameter estimates derived from the Capon algorithm degrades swiftly. Prediction of this threshold SNR point is of practical significance for robust system design and analysis. The exact pairwise error probabilities for the Capon (and Bartlett) algorithm, derived herein, are given by simple finite sums involving no numerical integration, include finite sample effects, and hold for an arbitrary colored data covariance. Via an adaptation of an interval error based method, these error probabilities, along with the local error MSE predictions of Vaidyanathan and Buckley, facilitate accurate prediction of the Capon threshold region MSE performance for an arbitrary number of well separated sources, circumventing the need for numerous Monte Carlo simulations. A large sample closed-form approximation for the Capon threshold SNR is provided for uniform linear arrays. A new, exact, two-point measure of the probability of resolution for the Capon algorithm, that includes the deleterious effects of signal model mismatch, is a serendipitous byproduct of this analysis that predicts the SNRs required for closely spaced sources to be mutually resolvable by the Capon algorithm. Last, a general strategy is provided for obtaining accurate MSE predictions that account for signal model mismatch.

PDF's, confidence regions, and relevant statistics for a class of sample covariance-based array processors

We add to the many results on sample covariance matrix (SCM) dependent array processors by (i) weakening the traditional assumption of Gaussian data and (ii) providing for a class of array processors additional performance measures that are of value in practice. The data matrix is assumed drawn from a class of multivariate elliptically contoured (MEC) distributions. The performance measures include the exact probability density functions (PDFs), confidence regions, and moments of the weight vector (matrix), beam response, and beamformer output of certain SCM-based (SCB) array processors. The array processors considered include the SCB: (i) maximum-likelihood (ML) signal vector estimator, (ii) linearly constrained minimum variance beamformer (LCMV), (iii) minimum variance distortionless response beamformer (MVDR), and (iv) generalized sidelobe canceller (GSC) implementation of the LCMV beamformer. It is shown that the exact joint PDFs for the weight vectors/matrices of the aforementioned SCB array processors are a linear transformation from a complex multivariate extension of the standardized t-distribution. The SCB beam responses are generalized t-distributed, and the PDFs of the SCB beamformer outputs are given by Kummers function. All but the beamformer outputs are shown to be completely invariant statistics over the class of MECs considered.

Mean-squared error and threshold SNR prediction of maximum-likelihood signal parameter estimation with estimated colored noise covariances

An interval error-based method (MIE) of predicting mean squared error (MSE) performance of maximum-likelihood estimators (MLEs) is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix; an intermediate step central to adaptive array detection and parameter estimation. The successful application of MIE requires good approximations of two quantities: 1) interval error probabilities and 2) asymptotic (SNRrarrinfin) local MSE performance of the MLE. Exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein, that in conjunction with the Union Bound provide accurate prediction of the required interval error probabilities. The Crameacuter-Rao Bound (CRB) often provides adequate prediction of the asymptotic local MSE performance of MLE. The signal parameters, however, are decoupled from the colored noise parameters in the Fisher Information Matrix for the deterministic signal model, rendering the CRB incapable of reflecting loss due to colored noise covariance estimation. A new modification of the CRB involving a complex central beta random variable different from, but analogous to the Reed, Mallett, and Brennan beta loss factor provides a working solution to this problem, facilitating MSE prediction well into the threshold region with remarkable accuracy

A note on non-Gaussian adaptive array detection and signal parameter estimation

Kellys (1986) generalized likelihood ratio test (GLRT) statistic is reexamined under a broad class of data distributions known as complex multivariate elliptically contoured (MEC), which include the complex Gaussian as a special case. We show that, mathematically, Kellys GLRT test statistic is again obtained when the data matrix is assumed to be MEC distributed. The maximum-likelihood (ML) estimate for the signal parameters-alias the sample-covariance-based (SCB) minimum variance distortionless response beamformer output and, in general, the SCB linearly constrained minimum variance beamformer output-is likewise shown to be the same. These results have significant robustness implications for adaptive detection/estimation/beamforming in non-Gaussian environments.

Derived PDF of maximum likelihood signal estimator which employs an estimated noise covariance

A probability density function (PDF) for the maximum likelihood (ML) signal vector estimator is derived when the estimator relies on a noise sample covariance matrix (SCM) for evaluation. By using a complex Wishart probabilistic model for the distribution of the SCM, it is shown that the PDF of the adaptive ML (AML) signal estimator (alias the SCM based minimum variance distortionless response (MVDR) beamformer output and, more generally, the SCM based linearly constrained minimum variance (LCMV) beamformer output) is, in general, the confluent hypergeometric function of a complex matrix argument known as Kummers function. The AML signal estimator remains unbiased but only asymptotically efficient; moreover, the AML signal estimator converges in distribution to the ML signal estimator (known noise covariance). When the sample size of the estimated noise covariance matrix is fixed, it is demonstrated that there exists a dynamic tradeoff between signal-to-noise ratio (SNR) and noise adaptivity as the dimensionality of the array data (number of adaptive degrees of freedom) is varied, suggesting the existence of an optimal array data dimension that will yield the best performance.

Bayesian bounds for matched-field parameter estimation

Matched-field methods concern estimation of source locations and/or ocean environmental parameters by exploiting full wave modeling of acoustic waveguide propagation. Because of the nonlinear parameter-dependence of the signal field, the estimate is subject to ambiguities and the sidelobe contribution often dominates the estimation error below a threshold signal-to-noise ratio (SNR). To study the matched-field performance, three Bayesian lower bounds on mean-square error are developed: the Bayesian Crame/spl acute/r-Rao bound (BCRB), the Weiss-Weinstein bound (WWB), and the Ziv-Zakai bound (ZZB). Particularly, for a multiple-frequency, multiple-snapshot random signal model, a closed-form minimum probability of error associated with the likelihood ratio test is derived, which facilitates error analysis in a wide scope of applications. Analysis and example simulations demonstrate that 1) unlike the local CRB, the BCRB is not achieved by the maximum likelihood estimate (MLE) even at high SNR if the local performance is not uniform across the prior parameter space; 2) the ZZB gives the closest MLE performance prediction at most SNR levels of practical interest; 3) the ZZB can also be used to determine the necessary number of independent snapshots achieving the asymptotic performance of the MLE at a given SNR; 4) incoherent frequency averaging, which is a popular multitone processing approach, reduces the peak sidelobe error but may not improve the overall performance due to the increased ambiguity baseline; and finally, 5) effects of adding additional parameters (e.g., environmental uncertainty) can be well predicted from the parameter coupling.

A Fresh Look at the Bayesian Bounds of the Weiss-Weinstein Family

Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss-Weinstein family. Among this family, we have Bayesian Cramer-Rao bound, the Bobrovsky-MayerWolf-Zakai bound, the Bayesian Bhattacharyya bound, the Bobrovsky-Zakai bound, the Reuven-Messer bound, and the Weiss-Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer-Wolf, and Zakai. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven-Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakai bound, and the Bayesian Cramer-Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem.

Statistics of adaptive nulling and use of the generalized eigenrelation (GER) for modeling inhomogeneities in adaptive processing

This paper examines the integrity of the generalized eigenrelation (GER), which is an approach to assessing performance in an adaptive processing context involving covariance estimation when the adaptive processors are subject to undernulled interference. The GER is a mathematical relation, which if satisfied, often facilitates closed-form analysis of adaptive processors employing estimated covariances subject to inhomogeneities. The goal of this paper is to determine what impact this constraint has on the integrity of the adaptive nulling process. In order to examine the impact of the GER constraint on adaptive nulling, we establish fundamental statistical convergence properties of an adaptive null for the sample covariance-based (SCB) minimum variance distortionless response (MVDR) beamformer. Novel exact expressions relating the mean and variance of an adaptive null of a homogeneously trained beamformer to the mean and variance of a nonhomogeneous trained beamformer are derived. In addition, it is shown that the Reed et al. (1974) result for required sample support can be highly inaccurate under nonhomogeneous conditions. Indeed, the required sample support can at times depend directly on the power of the undernulled interference.