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. a posteriori (MAP) optimal estimate that coincides with an optimal solution to the 1 optimization [10]. In the MMV case, the joint sparsity among different (uncorrelated) snapshots is utilized by assuming the same sparse prior for the signals at all snapshots [12]. Correlations between snapshots have also been studied in a recent paper [13]. One merit of SBI is its flexibility in modeling sparse signals that can not only promote the sparsity of its solution, e.g., in [11], but also exploit the possible structure of the signal to be recovered, e.g., in [14]. Since the Bayesian inference is a probabilistic method and based on heuristics to some extent, one shortcoming of SBI is that it offers fewer guarantees on the signal recovery accuracy as compared with, for example, 1 optimization.Recent advancements in array signal processing include compressive (CS-) MUSIC [15] and subspace-augmented (SA-) MUSIC [16]. They are combinations of the conventional MUSIC technique and recent CS methods with guaranteed support recovery performance and can outperform MUSIC and standard CS approaches. Though existing CS-based approaches have shown their improvements in DOA estimation, e.g., their success in the case of limited snapshots, there are still difficulties in practical situations where the true DOAs are not on the sampling grid. On one hand, a dense sampling grid is necessary for accurate DOA estimation to reduce the gap between the true DOA and its nearest grid point since the estimated DOAs are constrained on the grid. On the other hand, a dense sampling grid leads to a highly coherent matrix that violates the condition for the sparse signal recovery. We refer to the model adopted in the standard CS methods as an on-grid model hereafter in the sense that the estimated DOAs are constrained on the fixed grid.An off-grid model for DOA estimation is studied in [17] where the estimated DOAs are no longer constrained in the sampling grid set. The model takes into ac...

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 which exploit signal sparsity, work directly on continuous frequencies, and completely resolve the grid mismatch problem of previous compressed sensing methods. In this work 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 2,0 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 2,1 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.Index Terms-Atomic norm, compressed sensing, direction of arrival (DOA) estimation, joint sparse frequency recovery, multiple measurement vectors (MMVs).

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 0 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.

Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complexvalued. We study the popular recovery method of 1-regularized least squares or LASSO. While several studies have shown that LASSO provides desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to solve the complex-valued LASSO problem and obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP. Our theoretical results are concerned with the case of i.i.d. Gaussian sensing matrices. Simulations confirm that our results hold for a larger class of random matrices.

Abstract-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., 1 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.