Covariance Estimation from Compressive Data Partitions using a Projected Gradient-based Algorithm

Covariance matrix estimation techniques require high acquisition costs that challenge the sampling systems' storing and transmission capabilities. For this reason, various acquisition approaches have been developed to simultaneously sense and compress the relevant information of the signal using random projections. However, estimating the covariance matrix from the random projections is an ill-posed problem that requires further information about the data, such as sparsity, low rank, or stationary behavior. Furthermore, this approach fails using high compression ratios. Therefore, this paper proposes an algorithm based on the projected gradient method to recover a low-rank or Toeplitz approximation of the covariance matrix. The proposed algorithm divides the data into subsets projected onto different subspaces, assuming that each subset contains an approximation of the signal statistics, improving the inverse problem's condition. The error induced by this assumption is analytically derived along with the convergence guarantees of the proposed method. Extensive simulations show that the proposed algorithm can effectively recover the covariance matrix of hyperspectral images with high compression ratios (8-15% approx) in noisy scenarios. Additionally, simulations and theoretical results show that filtering the gradient reduces the estimator's error recovering up to twice the number of eigenvectors.Comment: submitted to IEEE Transactions on Image Processin

Multitarget Joint Delay and Doppler Shift Estimation in Bistatic Passive Radar

Bistatic passive radar (BPR) system does not transmit any electromagnetic signal unlike the active radar, but employs an existing Illuminator of opportunity (IO) in the environment, for instance, a broadcast station, to detect and track the targets of interest. Therefore, a BPR system is comprised of two channels. One is the reference channel that collects only the IO signal, and the other is the surveillance channel which is used to capture the targets\u27 reflected signals. When the IO signal reflected from multiple targets is captured in the surveillance channel (SC) then estimating the delays and Doppler shifts of all the observed targets is a challenging problem. For BPR system, the signal processing algorithms developed so far models the IO waveform as a deterministic process and discretizes the delays and Doppler shifts parameters. In this thesis, we deal with the problem of jointly estimating the delays and Doppler shifts of multiple targets in a BPR system (i.e., a two channel system) when the unknown IO signal is modeled as a correlated stochastic process. Unlike the previous work, we take all the delays and Doppler shifts as continuous-valued parameters to avoid straddle loss due to discretization and propose a computationally efficient Expectation-Maximization (EM) based algorithm that breaks up the complex multidimensional maximum likelihood optimization problem into multiple separate optimization problems. The EM algorithm jointly provides the estimates of all the delays and Doppler shifts of the targets along with the estimate of each target\u27s component signal in the SC and the estimate of the unknown IO signal. We also derive the Cramer-Rao lower bound for the considered multitarget estimation problem with stochastic IO signal. Numerical simulations are presented where we compare our proposed EM-based multi-target estimator with the widely used conventional cross correlation estimator under different multitarget environments

An Introduction to Wishart Matrix Moments

These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics, non-commutative algebra, geometry, multivariate functional and spectral analysis, and of course statistics and probability theory. As a result, most of the classical topics in random matrix theory are technical, and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specific context of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary, we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example, we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied, along with new spectral and trace concentration-type inequalities