The Rank of the Covariance Matrix of an Evanescent Field

Evanescent random fields arise as a component of the 2-D Wold decomposition of homogenous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem

Foundational principles for large scale inference: Illustrations through correlation mining

When can reliable inference be drawn in the "Big Data" context? This paper presents a framework for answering this fundamental question in the context of correlation mining, with implications for general large scale inference. In large scale data applications like genomics, connectomics, and eco-informatics the dataset is often variable-rich but sample-starved: a regime where the number $n$ of acquired samples (statistical replicates) is far fewer than the number $p$ of observed variables (genes, neurons, voxels, or chemical constituents). Much of recent work has focused on understanding the computational complexity of proposed methods for "Big Data." Sample complexity however has received relatively less attention, especially in the setting when the sample size $n$ is fixed, and the dimension $p$ grows without bound. To address this gap, we develop a unified statistical framework that explicitly quantifies the sample complexity of various inferential tasks. Sampling regimes can be divided into several categories: 1) the classical asymptotic regime where the variable dimension is fixed and the sample size goes to infinity; 2) the mixed asymptotic regime where both variable dimension and sample size go to infinity at comparable rates; 3) the purely high dimensional asymptotic regime where the variable dimension goes to infinity and the sample size is fixed. Each regime has its niche but only the latter regime applies to exa-scale data dimension. We illustrate this high dimensional framework for the problem of correlation mining, where it is the matrix of pairwise and partial correlations among the variables that are of interest. We demonstrate various regimes of correlation mining based on the unifying perspective of high dimensional learning rates and sample complexity for different structured covariance models and different inference tasks

Forward Looking Radar: Interference Modelling, Characterization, and Suppression

This research characterizes forward looking radar performance while noting differences with traditionally examined sidelooking radar. The target detection problem for forward looking radar is extremely difficult due to the severe, heterogeneous and range dependent ground clutter. Consequently, forward looking radar detection represents an important but overlooked topic because of the increased difficulty compared to sidelooking radar. This void must be filled since most fighter aircraft use forward looking radar, making this topic intensely interesting to the Air Force. After characterizing forward looking radar performance, basic radar concepts along with advanced adaptive interference suppression techniques improve the output Signal-to-Interference-plus-Noise Ratio (SINR) and target detection rates using fixed false alarm for linear arrays. However, target detection probabilities and output SINR do not improve enough. Although the methods considered are adaptive in azimuth and Doppler, effective range ambiguous clutter mitigation requires elevation adaptivity, a feature not offered by linear arrays. The research continues by examining planar arrays. Elevation adaptivity combined with azimuth and Doppler adaptivity allows suppressing range ambiguous clutter and significantly increasing output SINR, detection probability, and maximum detection range. Specifically, three-dimensional Space-Time Adaptive Processing (3D STAP) techniques with adaptivity in elevation, azimuth, and Doppler achieve detection probability improvements of over 10 dB in required input SINR compared to two-dimensional (2D) STAP processing. Additionally, 3D STAP improves detection probability versus input SINR curves over 30 dB when compared to 2D conventional processing techniques. As a result, forward looking radars using 3D STAP have the capacity to detect targets that conventi

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically ``as likely'' as the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix \mSigma_{0}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario ($T \geq M$) while Part 2 deals with the under-sampled scenario ($T \leq M$)

Static Background Removal in Vehicular Radar: Filtering in Azimuth-Elevation-Doppler Domain

A significant challenge in autonomous driving systems lies in image understanding within complex environments, particularly dense traffic scenarios. An effective solution to this challenge involves removing the background or static objects from the scene, so as to enhance the detection of moving targets as key component of improving overall system performance. In this paper, we present an efficient algorithm for background removal in automotive radar applications, specifically utilizing a frequency-modulated continuous wave (FMCW) radar. Our proposed algorithm follows a three-step approach, encompassing radar signal preprocessing, three-dimensional (3D) ego-motion estimation, and notch filter-based background removal in the azimuth-elevation-Doppler domain. To begin, we model the received signal of the FMCW multiple-input multiple-output (MIMO) radar and develop a signal processing framework for extracting four-dimensional (4D) point clouds. Subsequently, we introduce a robust 3D ego-motion estimation algorithm that accurately estimates radar ego-motion speed, accounting for Doppler ambiguity, by processing the point clouds. Additionally, our algorithm leverages the relationship between Doppler velocity, azimuth angle, elevation angle, and radar ego-motion speed to identify the spectrum belonging to background clutter. Subsequently, we employ notch filters to effectively filter out the background clutter. The performance of our algorithm is evaluated using both simulated data and extensive experiments with real-world data. The results demonstrate its effectiveness in efficiently removing background clutter and enhacing perception within complex environments. By offering a fast and computationally efficient solution, our approach effectively addresses challenges posed by non-homogeneous environments and real-time processing requirements