Asymptotic regime for impropriety tests of complex random vectors

Impropriety testing for complex-valued vector has been considered lately due to potential applications ranging from digital communications to complex media imaging. This paper provides new results for such tests in the asymptotic regime, i.e. when the vector dimension and sample size grow commensurately to infinity. The studied tests are based on invariant statistics named impropriety coefficients. Limiting distributions for these statistics are derived, together with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in the Gaussian case. This characterization in the asymptotic regime allows also to identify a phase transition in Roy's test with potential application in detection of complex-valued low-rank subspace corrupted by proper noise in large datasets. Simulations illustrate the accuracy of the proposed asymptotic approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS

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

Performance of Statistical Tests for Single Source Detection using Random Matrix Theory

This paper introduces a unified framework for the detection of a source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver. The Generalized Maximum Likelihood Test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace. Using recent results of random matrix theory, a practical way to evaluate the threshold and the $p$-value of the test is provided in the asymptotic regime where the number $K$ of sensors and the number $N$ of observations per sensor are large but have the same order of magnitude. The theoretical performance of the test is then analyzed in terms of Receiver Operating Characteristic (ROC) curve. It is in particular proved that both Type I and Type II error probabilities converge to zero exponentially as the dimensions increase at the same rate, and closed-form expressions are provided for the error exponents. These theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models, and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix.Comment: 45 p. improved presentation; more proofs provide

Matched direction detectors and estimators for array processing with subspace steering vector uncertainties

In this paper, we consider the problem of estimating and detecting a signal whose associated spatial signature is known to lie in a given linear subspace but whose coordinates in this subspace are otherwise unknown, in the presence of subspace interference and broad-band noise. This situation arises when, on one hand, there exist uncertainties about the steering vector but, on the other hand, some knowledge about the steering vector errors is available. First, we derive the maximum-likelihood estimator (MLE) for the problem and compute the corresponding Cramer-Rao bound. Next, the maximum-likelihood estimates are used to derive a generalized likelihood ratio test (GLRT). The GLRT is compared and contrasted with the standard matched subspace detectors. The performances of the estimators and detectors are illustrated by means of numerical simulations

On the influence of detection tests on deterministic parameters estimation

In non-linear estimation problems three distinct regions of operation can be observed. In the asymptotic region, the Mean Square Error (MSE) of Maximum Likelihood Estimators (MLE) is small and, in many cases,close to the Cramer-Rao bound (CRB). In the a priory performance region where the number of independent snapshots and/or the SNR are very low, the MSE is close to that obtained from the prior knowledge about the problem. Between these two extremes, there is an additional transition region where MSE of estimators deteriorates with respect to CRB. The present paper provides exemples of improvement of MSE prediction by CRB, not only in the transition region but also in the a priori region, resulting from introduction of a detection step, which proves that this renement in MSE lower bounds derivation is worth investigating

Signal Processing in Large Systems: a New Paradigm

For a long time, detection and parameter estimation methods for signal processing have relied on asymptotic statistics as the number $n$ of observations of a population grows large comparatively to the population size $N$, i.e. $n/N\to \infty$. Modern technological and societal advances now demand the study of sometimes extremely large populations and simultaneously require fast signal processing due to accelerated system dynamics. This results in not-so-large practical ratios $n/N$, sometimes even smaller than one. A disruptive change in classical signal processing methods has therefore been initiated in the past ten years, mostly spurred by the field of large dimensional random matrix theory. The early works in random matrix theory for signal processing applications are however scarce and highly technical. This tutorial provides an accessible methodological introduction to the modern tools of random matrix theory and to the signal processing methods derived from them, with an emphasis on simple illustrative examples

Asymptotic Signal Detection Rates with 1-bit Array Measurements

This work considers detecting the presence of a band-limited random radio source using an antenna array featuring a low-complexity digitization process with single-bit output resolution. In contrast to high-resolution analog-to-digital conversion, such a direct transformation of the analog radio measurements to a binary representation can be implemented hardware and energy-efficient. However, the probabilistic model of the binary receive data becomes challenging. Therefore, we first consider the Neyman-Pearson test within generic exponential families and derive the associated analytic detection rate expressions. Then we use a specific replacement model for the binary likelihood and study the achievable detection performance with 1- bit radio array measurements. As an application, we explore the capability of a low-complexity GPS spectrum monitoring system with different numbers of antennas and different observation intervals. Results show that with a moderate amount of binary sensors it is possible to reliably perform the monitoring task

Biologically Inspired Sensing and MIMO Radar Array Processing

The contributions of this dissertation are in the fields of biologically inspired sensing and multi-input multi-output: MIMO) radar array processing. In our research on biologically inspired sensing, we focus on the mechanically coupled ears of the female Ormia ochracea. Despite the small distance between its ears, the Ormia has a remarkable localization ability. We statistically analyze the localization accuracy of the Ormia\u27s coupled ears, and illustrate the improvement in the localization performance due to the mechanical coupling. Inspired by the Ormia\u27s ears, we analytically design coupled small-sized antenna arrays with high localization accuracy and radiation performance. Such arrays are essential for sensing systems in military and civil applications, which are confined to small spaces. We quantitatively demonstrate the improvement in the antenna array\u27s radiation and localization performance due to the biologically inspired coupling. On MIMO radar, we first propose a statistical target detection method in the presence of realistic clutter. We use a compound-Gaussian distribution to model the heavy tailed characteristics of sea and foliage clutter. We show that MIMO radars are useful to discriminate a target from clutter using the spatial diversity of the illuminated area, and hence MIMO radar outperforms conventional phased-array radar in terms of target-detection capability. Next, we develop a robust target detector for MIMO radar in the presence of a phase synchronization mismatch between transmitter and receiver pairs. Such mismatch often occurs due to imperfect knowledge of the locations as well as local oscillator characteristics of the antennas, but this fact has been ignored by most researchers. Considering such errors, we demonstrate the degradation in detection performance. Finally, we analyze the sensitivity of MIMO radar target detection to changes in the cross-correlation levels: CCLs) of the received signals. Prior research about MIMO radar assumes orthogonality among the received signals for all delay and Doppler pairs. However, due to the use of antennas which are widely separated in space, it is impossible to maintain this orthogonality in practice. We develop a target-detection method considering the non-orthogonality of the received data. In contrast to the common assumption, we observe that the effect of non-orthogonality is significant on detection performance

Space-time reduced rank methods and CFAR signal detection algorithms with applications to HPRF radar

In radar applications, the statistical properties (covariance matrix) of the interference are typically unknown a priori and are estimated from a dataset with limited sample support. Often, the limited sample support leads to numerically ill-conditioned radar detectors. Under such circumstances, classical interference cancellation methods such as sample matrix inversion (SMI) do not perform satisfactorily. In these cases, innovative reduced-rank space-time adaptive processing (STAP) techniques outperform full-rank techniques. The high pulse repetition frequency (HPRF) radar problem is analyzed and it is shown that it is in the class of adaptive radar with limited sample support. Reduced-rank methods are studied for the HPRF radar problem. In particular, the method known as diagonally loaded covariance matrix SMI (L-SMI) is closely investigated. Diagonal loading improves the numerical conditioning of the estimated covariance matrix, and hence, is well suited to be applied in a limited sample support environment. The performance of L-SMI is obtained through a theoretical distribution of the output conditioned signal-to-noise ratio of the space-time array. Reduced-rank techniques are extended to constant false alarm rate (CFAR) detectors based on the generalized likelihood ratio test (GLRT). Two new modified CFAR GLRT detectors are considered and analyzed. The first is a subspace-based GLRT detector where subspace-based transformations are applied to the data prior to detection. A subspace transformation adds statistical stability which tends to improve performance at the expense of an additional SNR loss. The second detector is a modified GLRT detector that incorporates a diagonally loaded covariance matrix. Both detectors show improved performance over the traditional GLRT