Passive detection of rank-one signals with a multiantenna reference channel

In this work we consider a two-channel passive detection problem, in which there is a surveillance array where the presence/absence of a target signal is to be detected, and a reference array that provides a noise-contaminated version of the target signal. We assume that the transmitted signal is an unknown rank-one signal, and that the noises are uncorrelated between the two channels, but each one having an unknown and arbitrary spatial covariance matrix. We show that the generalized likelihood ratio test (GLRT) for this problem rejects the null hypothesis when the largest canonical correlation of the sample coherence matrix between the surveillance and the reference channels exceeds a threshold. Further, based on recent results from random matrix theory, we provide an approximation for the null distribution of the test statistic.The work of I. Santamaría was supported by the Spanish Government through grants PRX14/0028 (Estancias de Movilidad de Profesores, Ministerio de Educación) and by project RACHEL (TEC2013-47141-C4-3-R) funded by the Ministerio de Economía y Competitividad (MINECO). The work of L. Scharf and D. Cochran was supported in part by a sub-contract with Matrix Research for research sponsored by the Air Force Research Laboratory under contract FA8650-14-D-1722

Analysis and Design of Multiple-Antenna Cognitive Radios with Multiple Primary User Signals

We consider multiple-antenna signal detection of primary user transmission signals by a secondary user receiver in cognitive radio networks. The optimal detector is analyzed for the scenario where the number of primary user signals is no less than the number of receive antennas at the secondary user. We first derive exact expressions for the moments of the generalized likelihood ratio test (GLRT) statistic, yielding approximations for the false alarm and detection probabilities. We then show that the normalized GLRT statistic converges in distribution to a Gaussian random variable when the number of antennas and observations grow large at the same rate. Further, using results from large random matrix theory, we derive expressions to compute the detection probability without explicit knowledge of the channel, and then particularize these expressions for two scenarios of practical interest: 1) a single primary user sending spatially multiplexed signals, and 2) multiple spatially distributed primary users. Our analytical results are finally used to obtain simple design rules for the signal detection threshold.Comment: Revised version (14 pages). Change in titl

Numerical Computation of Wishart Eigenvalue Distributions for Multistatic Radar Detection

abstract: Eigenvalues of the Gram matrix formed from received data frequently appear in sufficient detection statistics for multi-channel detection with Generalized Likelihood Ratio (GLRT) and Bayesian tests. In a frequently presented model for passive radar, in which the null hypothesis is that the channels are independent and contain only complex white Gaussian noise and the alternative hypothesis is that the channels contain a common rank-one signal in the mean, the GLRT statistic is the largest eigenvalue $\lambda_1$ of the Gram matrix formed from data. This Gram matrix has a Wishart distribution. Although exact expressions for the distribution of $\lambda_1$ are known under both hypotheses, numerically calculating values of these distribution functions presents difficulties in cases where the dimension of the data vectors is large. This dissertation presents tractable methods for computing the distribution of $\lambda_1$ under both the null and alternative hypotheses through a technique of expanding known expressions for the distribution of $\lambda_1$ as inner products of orthogonal polynomials. These newly presented expressions for the distribution allow for computation of detection thresholds and receiver operating characteristic curves to arbitrary precision in floating point arithmetic. This represents a significant advancement over the state of the art in a problem that could previously only be addressed by Monte Carlo methods.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

Informative Data Fusion: Beyond Canonical Correlation Analysis

Multi-modal data fusion is a challenging but common problem arising in fields such as economics, statistical signal processing, medical imaging, and machine learning. In such applications, we have access to multiple datasets that use different data modalities to describe some system feature. Canonical correlation analysis (CCA) is a multidimensional joint dimensionality reduction algorithm for exactly two datasets. CCA finds a linear transformation for each feature vector set such that the correlation between the two transformed feature sets is maximized. These linear transformations are easily found by solving the SVD of a matrix that only involves the covariance and cross-covariance matrices of the feature vector sets. When these covariance matrices are unknown, an empirical version of CCA substitutes sample covariance estimates formed from training data. However, when the number of training samples is less than the combined dimension of the datasets, CCA fails to reliably detect correlation between the datasets. This thesis explores the the problem of detecting correlations from data modeled by the ubiquitous signal-plus noise data model. We present a modification to CCA, which we call informative CCA (ICCA) that first projects each dataset onto a low-dimensional informative signal subspace. We verify the superior performance of ICCA on real-world datasets and argue the optimality of trim-then-fuse over fuse-then-trim correlation analysis strategies. We provide a significance test for the correlations returned by ICCA and derive improved estimates of the population canonical vectors using insights from random matrix theory. We then extend the analysis of CCA to regularized CCA (RCCA) and demonstrate that setting the regularization parameter to infinity results in the best performance and has the same solution as taking the SVD of the cross-covariance matrix of the two datasets. Finally, we apply the ideas learned from ICCA to multiset CCA (MCCA), which analyzes correlations for more than two datasets. There are multiple formulations of multiset CCA (MCCA), each using a different combination of objective function and constraint function to describe a notion of multiset correlation. We consider MAXVAR, provide an informative version of the algorithm, which we call informative MCCA (IMCCA), and demonstrate its superiority on a real-world dataset.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113419/1/asendorf_1.pd