Performance and Robustness Analysis of Co-Prime and Nested Sampling

Coprime and nested sampling are well known deterministic sampling techniques that operate at rates significantly lower than the Nyquist rate, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This thesis studies the performance of coprime and nested samplers in spatial and temporal domains, when these assumptions are violated. In spatial domain, the robustness of these samplers is studied by considering arrays with perturbed sensor locations (with unknown perturbations). Simplified expressions for the Fisher Information matrix for perturbed coprime and nested arrays are derived, which explicitly highlight the role of co-array. It is shown that even in presence of perturbations, it is possible to resolve $O(M^2)$ under appropriate conditions on the size of the grid. The assumption of small perturbations leads to a novel ``bi-affine" model in terms of source powers and perturbations. The redundancies in the co-array are then exploited to eliminate the nuisance perturbation variable, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers. This thesis also studies the robustness of coprime sampling to finite number of samples and sampling jitter, by analyzing their effects on the quality of the estimated autocorrelation sequence. A variety of bounds on the error introduced by such non ideal sampling schemes are computed by considering a statistical model for the perturbation. They indicate that coprime sampling leads to stable estimation of the autocorrelation sequence, in presence of small perturbations. Under appropriate assumptions on the distribution of WSS signals, sharp bounds on the estimation error are established which indicate that the error decays exponentially with the number of samples. The theoretical claims are supported by extensive numerical experiments

DOA Estimation in Partially Correlated Noise Using Low-Rank/Sparse Matrix Decomposition

We consider the problem of direction-of-arrival (DOA) estimation in unknown partially correlated noise environments where the noise covariance matrix is sparse. A sparse noise covariance matrix is a common model for a sparse array of sensors consisted of several widely separated subarrays. Since interelement spacing among sensors in a subarray is small, the noise in the subarray is in general spatially correlated, while, due to large distances between subarrays, the noise between them is uncorrelated. Consequently, the noise covariance matrix of such an array has a block diagonal structure which is indeed sparse. Moreover, in an ordinary nonsparse array, because of small distance between adjacent sensors, there is noise coupling between neighboring sensors, whereas one can assume that nonadjacent sensors have spatially uncorrelated noise which makes again the array noise covariance matrix sparse. Utilizing some recently available tools in low-rank/sparse matrix decomposition, matrix completion, and sparse representation, we propose a novel method which can resolve possibly correlated or even coherent sources in the aforementioned partly correlated noise. In particular, when the sources are uncorrelated, our approach involves solving a second-order cone programming (SOCP), and if they are correlated or coherent, one needs to solve a computationally harder convex program. We demonstrate the effectiveness of the proposed algorithm by numerical simulations and comparison to the Cramer-Rao bound (CRB).Comment: in IEEE Sensor Array and Multichannel signal processing workshop (SAM), 201

Successive Concave Sparsity Approximation for Compressed Sensing

In this paper, based on a successively accuracy-increasing approximation of the $\ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $\ell_0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $\ell_1$ and $\ell_0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted $\ell_1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.Comment: Submitted to IEEE Trans. on Signal Processin

Sampling for Underdetermined Linear and Multilinear Inverse Problems: role of geometry and statistical priors

Estimating the underlying parameters of a statistical signal from noisy observations is a central problem in signal processing, with a wide variety of applications in many different fields such as machine learning, source localization, channel estimation for modern millimeter-wave (mmWave) communication systems, etc. Classical algorithms for source localization guarantee recovery of only $K=\mathcal{O}(M)$ sources using a Uniform Linear Array equipped with $M$ antennas. Recently, it has been shown that using certain non-uniform array designs, such as coprime and nested arrays, once certain correlation priors are assumed, it is possible to break this limit and go all the way up to $K=\mathcal{O}(M^2)$ sources. This thesis sheds more light on this phenomena, and more general cases of under-determined inverse problems in both linear, and non-linear settings. We show that for linear inverse problems, not only CRB exists for the case that $K=\mathcal{O}(M^2)$, for certain non-uniform arrays, but also it continues to exist even if the antenna locations are perturbed due to physical deformation of the device, or only a compressed version of the measurements are available. For a more general class of linear inverse problems, we show that in presence of certain correlation priors, one can recover sparse vectors of sparsity $K=\mathcal{O}(M^2)$, where the probability of detecting a wrong support for the sparse vector decays to zero exponentially fast as more and more temporal snapshots are obtained. We show these results hold for a variety of different statistical models, namely Gaussian sources, bounded sources, when the measurement matrices are equi-angular tight frames, and finally for the case that the measurements are obtained in adaptively. This thesis also considers multilinear inverse problems, namely tensor decompositions as well as certain non-convex problems with applications in millimeter-wave (mmWave) communication systems. We propose tensor decomposition algorithms for channel estimation for mmWave communication systems equipped with hybrid analog/digital beamforming, for cases such as multi-carrier Single-Input/Multiple-Output and single-carrier Multiple-Input/Multiple-Output, where we show the immense benefit gained by posing certain commonly considered statistical assumptions on the channel parameters, which leads to a provable increased identifiability compared to the existing algorithms for mmWave channel estimation

Sampling for Underdetermined Linear and Multilinear Inverse Problems: role of geometry and statistical priors

Estimating the underlying parameters of a statistical signal from noisy observations is a central problem in signal processing, with a wide variety of applications in many different fields such as machine learning, source localization, channel estimation for modern millimeter-wave (mmWave) communication systems, etc. Classical algorithms for source localization guarantee recovery of only $K=\mathcal{O}(M)$ sources using a Uniform Linear Array equipped with $M$ antennas. Recently, it has been shown that using certain non-uniform array designs, such as coprime and nested arrays, once certain correlation priors are assumed, it is possible to break this limit and go all the way up to $K=\mathcal{O}(M^2)$ sources. This thesis sheds more light on this phenomena, and more general cases of under-determined inverse problems in both linear, and non-linear settings. We show that for linear inverse problems, not only CRB exists for the case that $K=\mathcal{O}(M^2)$, for certain non-uniform arrays, but also it continues to exist even if the antenna locations are perturbed due to physical deformation of the device, or only a compressed version of the measurements are available. For a more general class of linear inverse problems, we show that in presence of certain correlation priors, one can recover sparse vectors of sparsity $K=\mathcal{O}(M^2)$, where the probability of detecting a wrong support for the sparse vector decays to zero exponentially fast as more and more temporal snapshots are obtained. We show these results hold for a variety of different statistical models, namely Gaussian sources, bounded sources, when the measurement matrices are equi-angular tight frames, and finally for the case that the measurements are obtained in adaptively. This thesis also considers multilinear inverse problems, namely tensor decompositions as well as certain non-convex problems with applications in millimeter-wave (mmWave) communication systems. We propose tensor decomposition algorithms for channel estimation for mmWave communication systems equipped with hybrid analog/digital beamforming, for cases such as multi-carrier Single-Input/Multiple-Output and single-carrier Multiple-Input/Multiple-Output, where we show the immense benefit gained by posing certain commonly considered statistical assumptions on the channel parameters, which leads to a provable increased identifiability compared to the existing algorithms for mmWave channel estimation