Zai Yang

Off-Grid Direction of Arrival Estimation Using Sparse Bayesian Inference

Direction of arrival (DOA) estimation is a classical problem in signal processing with many practical applications. Its research has recently been advanced owing to the development of methods based on sparse signal reconstruction. While these methods have shown advantages over conventional ones, there are still difficulties in practical situations where true DOAs are not on the discretized sampling grid. To deal with such an off-grid DOA estimation problem, this paper studies an off-grid model that takes into account effects of the off-grid DOAs and has a smaller modeling error. An iterative algorithm is developed based on the off-grid model from a Bayesian perspective while joint sparsity among different snapshots is exploited by assuming a Laplace prior for signals at all snapshots. The new approach applies to both single snapshot and multi-snapshot cases. Numerical simulations show that the proposed algorithm has improved accuracy in terms of mean squared estimation error. The algorithm can maintain high estimation accuracy even under a very coarse sampling grid.

Robustly Stable Signal Recovery in Compressed Sensing With Structured Matrix Perturbation

The sparse signal recovery in the standard compressed sensing (CS) problem requires that the sensing matrix be known a priori. Such an ideal assumption may not be met in practical applications where various errors and fluctuations exist in the sensing instruments. This paper considers the problem of compressed sensing subject to a structured perturbation in the sensing matrix. Under mild conditions, it is shown that a sparse signal can be recovered by l1 minimization and the recovery error is at most proportional to the measurement noise level, which is similar to the standard CS result. In the special noise free case, the recovery is exact provided that the signal is sufficiently sparse with respect to the perturbation level. The formulated structured sensing matrix perturbation is applicable to the direction of arrival estimation problem, so has practical relevance. Algorithms are proposed to implement the l1 minimization problem and numerical simulations are carried out to verify the results obtained.

On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data

This paper is concerned about sparse, continuous frequency estimation in line spectral estimation, and focused on developing gridless sparse methods which overcome grid mismatches and correspond to limiting scenarios of existing grid-based approaches, e.g., ℓ1 optimization and SPICE, with an infinitely dense grid. We generalize AST (atomic-norm soft thresholding) to the case of nonconsecutively sampled data (incomplete data) inspired by recent atomic norm based techniques. We present a gridless version of SPICE (gridless SPICE, or GLS), which is applicable to both complete and incomplete data without the knowledge of noise level. We further prove the equivalence between GLS and atomic norm-based techniques under different assumptions of noise. Moreover, we extend GLS to a systematic framework consisting of model order selection and robust frequency estimation, and present feasible algorithms for AST and GLS. Numerical simulations are provided to validate our theoretical analysis and demonstrate performance of our methods compared to existing ones.

Exact Joint Sparse Frequency Recovery via Optimization Methods

Frequency recovery/estimation from discrete samples of superimposed sinusoidal signals is a classic yet important problem in statistical signal processing. Its research has recently been advanced by atomic norm techniques that exploit signal sparsity, work directly on continuous frequencies, and completely resolve the grid mismatch problem of previous compressed sensing methods. In this paper, we investigate the frequency recovery problem in the presence of multiple measurement vectors (MMVs) which share the same frequency components, termed as joint sparse frequency recovery and arising naturally from array processing applications. To study the advantage of MMVs, we first propose an ℓ2,0 norm like approach by exploiting joint sparsity and show that the number of recoverable frequencies can be increased except in a trivial case. While the resulting optimization problem is shown to be rank minimization that cannot be practically solved, we then propose an MMV atomic norm approach that is a convex relaxation and can be viewed as a continuous counterpart of the ℓ2,1 norm method. We show that this MMV atomic norm approach can be solved by semidefinite programming. We also provide theoretical results showing that the frequencies can be exactly recovered under appropriate conditions. The above results either extend the MMV compressed sensing results from the discrete to the continuous setting or extend the recent super-resolution and continuous compressed sensing framework from the single to the multiple measurement vectors case. Extensive simulation results are provided to validate our theoretical findings and they also imply that the proposed MMV atomic norm approach can improve the performance in terms of reduced number of required measurements and/or relaxed frequency separation condition.

A discretization-free sparse and parametric approach for linear array signal processing

Direction of arrival (DOA) estimation in array processing using uniform/sparse linear arrays is concerned in this paper. While sparse methods via approximate parameter discretization have been popular in the past decade, the discretization may cause problems, e.g., modeling error and increased computations due to dense sampling. In this paper, an exact discretization-free method, named as sparse and parametric approach (SPA), is proposed for uniform and sparse linear arrays. SPA carries out parameter estimation in the continuous range based on well-established covariance fitting criteria and convex optimization. It guarantees to produce a sparse parameter estimate without discretization required by existing sparse methods. Theoretical analysis shows that the SPA parameter estimator is a large-snapshot realization of the maximum likelihood estimator and is statistically consistent (in the number of snapshots) under uncorrelated sources. Other merits of SPA include improved resolution, applicability to arbitrary number of snapshots, robustness to correlation of the sources and no requirement of user-parameters. Numerical simulations are carried out to verify our analysis and demonstrate advantages of SPA compared to existing methods.

Enhancing Sparsity and Resolution via Reweighted Atomic Norm Minimization

The mathematical theory of super-resolution developed recently by Candès and Fernandes-Granda states that a continuous, sparse frequency spectrum can be recovered with infinite precision via a (convex) atomic norm technique given a set of uniform time-space samples. This theory was then extended to the cases of partial/compressive samples and/or multiple measurement vectors via atomic norm minimization (ANM), known as off-grid/continuous compressed sensing (CCS). However, a major problem of existing atomic norm methods is that the frequencies can be recovered only if they are sufficiently separated, prohibiting commonly known high resolution. In this paper, a novel (nonconvex) sparse metric is proposed that promotes sparsity to a greater extent than the atomic norm. Using this metric an optimization problem is formulated and a locally convergent iterative algorithm is implemented. The algorithm iteratively carries out ANM with a sound reweighting strategy which enhances sparsity and resolution, and is termed as reweighted atomic-norm minimization (RAM). Extensive numerical simulations are carried out to demonstrate the advantageous performance of RAM with application to direction of arrival (DOA) estimation.

TDOA-Based Source Localization With Distance-Dependent Noises

This paper focuses on the problem of source localization using time-difference-of-arrival (TDOA) measurements in both 2-D and 3-D spaces. Different from existing studies where the variance of TDOA measurement noises is assumed to be independent of the associated source-to-sensor distances, we consider the more realistic model where the variance is a function of the source-to-sensor distances, which dramatically complicates TDOA-based source localization. After formulating the distance-dependent noise model, we prove that using the extra information about the source location in the functional variance improves the estimation accuracy of TDOA-based source localization, but contributes little under a sufficiently small noise level. Further, we theoretically analyze the problem of optimal sensor placement, and derive the necessary and sufficient conditions for optimizing localization performance under different circumstances. Then, a localization scheme based on the iteratively reweighted generalized least squares (IRGLS) method is proposed to efficiently exploit the extra source location information. Finally, a simulation analysis confirms our theoretical studies, and shows that the performance of the proposed localization scheme is comparable to the Cramer-Rao lower bound (CRLB) given moderate TDOA measurement noises.

Vandermonde Decomposition of Multilevel Toeplitz Matrices With Application to Multidimensional Super-Resolution

The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1-D frequency estimation. Many related numerical tools have also been developed for multidimensional (MD), especially 2-D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds for multilevel Toeplitz matrices in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided when the matrix rank is lower than the dimension of each Toeplitz block. A numerical method for searching for a decomposition is also proposed when the matrix rank is higher. The new results are applied to study the MD frequency estimation within the recent super-resolution framework. A precise formulation of the atomic

Continuous compressed sensing with a single or multiple measurement vectors

\ell _{0}

Orthonormal Expansion

norm is derived using the Vandermonde decomposition. Practical algorithms for frequency estimation are proposed based on the relaxation techniques. Extensive numerical simulations are provided to demonstrate the effectiveness of these algorithms compared with the existing atomic norm and subspace methods.