Sparsity-Cognizant Total Least-Squares for Perturbed Compressive Sampling

Solving linear regression problems based on the total least-squares (TLS) criterion has well-documented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. However, existing TLS approaches do not account for sparsity possibly present in the unknown vector of regression coefficients. On the other hand, sparsity is the key attribute exploited by modern compressive sampling and variable selection approaches to linear regression, which include noise in the data, but do not account for perturbations in the regression matrix. The present paper fills this gap by formulating and solving TLS optimization problems under sparsity constraints. Near-optimum and reduced-complexity suboptimum sparse (S-) TLS algorithms are developed to address the perturbed compressive sampling (and the related dictionary learning) challenge, when there is a mismatch between the true and adopted bases over which the unknown vector is sparse. The novel S-TLS schemes also allow for perturbations in the regression matrix of the least-absolute selection and shrinkage selection operator (Lasso), and endow TLS approaches with ability to cope with sparse, under-determined "errors-in-variables" models. Interesting generalizations can further exploit prior knowledge on the perturbations to obtain novel weighted and structured S-TLS solvers. Analysis and simulations demonstrate the practical impact of S-TLS in calibrating the mismatch effects of contemporary grid-based approaches to cognitive radio sensing, and robust direction-of-arrival estimation using antenna arrays.Comment: 30 pages, 10 figures, submitted to IEEE Transactions on Signal Processin

Multiple and single snapshot compressive beamforming

For a sound field observed on a sensor array, compressive sensing (CS) reconstructs the direction-of-arrival (DOA) of multiple sources using a sparsity constraint. The DOA estimation is posed as an underdetermined problem by expressing the acoustic pressure at each sensor as a phase-lagged superposition of source amplitudes at all hypothetical DOAs. Regularizing with an $\ell_1$-norm constraint renders the problem solvable with convex optimization, and promoting sparsity gives high-resolution DOA maps. Here, the sparse source distribution is derived using maximum a posteriori (MAP) estimates for both single and multiple snapshots. CS does not require inversion of the data covariance matrix and thus works well even for a single snapshot where it gives higher resolution than conventional beamforming. For multiple snapshots, CS outperforms conventional high-resolution methods, even with coherent arrivals and at low signal-to-noise ratio. The superior resolution of CS is demonstrated with vertical array data from the SWellEx96 experiment for coherent multi-paths.Comment: In press Journal of Acoustical Society of Americ

A Compact Formulation for the ℓ2,1\ell_{2,1} Mixed-Norm Minimization Problem

Parameter estimation from multiple measurement vectors (MMVs) is a fundamental problem in many signal processing applications, e.g., spectral analysis and direction-of- arrival estimation. Recently, this problem has been address using prior information in form of a jointly sparse signal structure. A prominent approach for exploiting joint sparsity considers mixed-norm minimization in which, however, the problem size grows with the number of measurements and the desired resolution, respectively. In this work we derive an equivalent, compact reformulation of the $\ell_{2,1}$ mixed-norm minimization problem which provides new insights on the relation between different existing approaches for jointly sparse signal reconstruction. The reformulation builds upon a compact parameterization, which models the row-norms of the sparse signal representation as parameters of interest, resulting in a significant reduction of the MMV problem size. Given the sparse vector of row-norms, the jointly sparse signal can be computed from the MMVs in closed form. For the special case of uniform linear sampling, we present an extension of the compact formulation for gridless parameter estimation by means of semidefinite programming. Furthermore, we derive in this case from our compact problem formulation the exact equivalence between the $\ell_{2,1}$ mixed-norm minimization and the atomic-norm minimization. Additionally, for the case of irregular sampling or a large number of samples, we present a low complexity, grid-based implementation based on the coordinate descent method

Robust Target Positioning for Reconfigurable Intelligent Surface Assisted MIMO Radar Systems

The direction of arrival (DOA) based multiple-input multiple-output (MIMO) radar technique has been widely utilized for ubiquitous positioning due to its advantage of simple implementability. On the other hand, reconfigurable intelligent surface (RIS) has received considerable attention, which can be deployed on the walls and objects to strengthen the positioning performance. However, RIS is usually not equipped with a perception module, which results in the tremendous challenge for RIS-assisted positioning. To tackle this challenge, this paper propose the fundamental problem of DOA-based target positioning in RIS-assisted MIMO radar system. Unlike conventional DOA estimation systems, the beneficial role of RIS is investigated in MIMO radar system, where a nonconvex $\ell _{p}$ promoting function is exploited to estimate DOA task. By adjusting the reflecting elements of the RIS, the proximal projection iterative strategy is developed to obtain the feasible solution. Both theoretical analysis and simulation results illustrate that the proposed scheme can achieve remarkable positioning performance and shed light on the benefits offered by the adoption of the RIS in terms of positioning performance

Space Time MUSIC: Consistent Signal Subspace Estimation for Wide-band Sensor Arrays

Wide-band Direction of Arrival (DOA) estimation with sensor arrays is an essential task in sonar, radar, acoustics, biomedical and multimedia applications. Many state of the art wide-band DOA estimators coherently process frequency binned array outputs by approximate Maximum Likelihood, Weighted Subspace Fitting or focusing techniques. This paper shows that bin signals obtained by filter-bank approaches do not obey the finite rank narrow-band array model, because spectral leakage and the change of the array response with frequency within the bin create \emph{ghost sources} dependent on the particular realization of the source process. Therefore, existing DOA estimators based on binning cannot claim consistency even with the perfect knowledge of the array response. In this work, a more realistic array model with a finite length of the sensor impulse responses is assumed, which still has finite rank under a space-time formulation. It is shown that signal subspaces at arbitrary frequencies can be consistently recovered under mild conditions by applying MUSIC-type (ST-MUSIC) estimators to the dominant eigenvectors of the wide-band space-time sensor cross-correlation matrix. A novel Maximum Likelihood based ST-MUSIC subspace estimate is developed in order to recover consistency. The number of sources active at each frequency are estimated by Information Theoretic Criteria. The sample ST-MUSIC subspaces can be fed to any subspace fitting DOA estimator at single or multiple frequencies. Simulations confirm that the new technique clearly outperforms binning approaches at sufficiently high signal to noise ratio, when model mismatches exceed the noise floor.Comment: 15 pages, 10 figures. Accepted in a revised form by the IEEE Trans. on Signal Processing on 12 February 1918. @IEEE201

Sparsity of the Field Signal-Based Method for Improving Spatial Resolution in Antenna Sensor Array Processing

The goal of array processing is to gather information from propagating radio-wave signals, as their Direction Of Arrival (DOA). The estimation of the DOA can be carried out by extracting the information of interest from the steering vector relevant to the adopted antenna sensor array. Such task can be accomplished in a number of different ways. However, in source estimation problems, it is essential to make use of a processing algorithm which feature not only good accuracy under ideal working conditions, but also robustness against non-idealities such as noise, limitations in the amount of collectible data, correlation between the sources, and modeling errors. In this work particular attention is devoted to spectrum estimation approaches based on sparsity. Conventional algorithms based on Beamforming fail wherein the radio sources are not within Rayleigh resolution range which is a function of the number of sensors and the dimension of the array. DOA estimation techniques such as MUSIC (MUltiple Signal Classifications) allow having a larger spatial resolution compared to Beamforming-based procedures, but if the sources are very close and the Signal to Noise Ratio (SNR) level is low, the resolution turns to be low as well. A better resolution can be obtained by exploiting sparsity: if the number of sources is small, the power spectrum of the signal with respect to the location is sparse. In this way, sparsity can enhance the accuracy of the estimation. In this paper, an estimation procedure based on the sparsity of the radio signals and useful to improve the conventional MUSIC method is presented and analyzed. The sparsity level is set in order to focus the signal energy only along the actual direction of arrival. The obtained numerical results have shown an improvement of the spatial resolution as well as a reduced error in DOA estimation with respect to conventional techniques

Augmented Lagrange Based on Modified Covariance Matching Criterion Method for DOA Estimation in Compressed Sensing

A novel direction of arrival (DOA) estimation method in compressed sensing (CS) is presented, in which DOA estimation is considered as the joint sparse recovery from multiple measurement vectors (MMV). The proposed method is obtained by minimizing the modified-based covariance matching criterion, which is acquired by adding penalties according to the regularization method. This minimization problem is shown to be a semidefinite program (SDP) and transformed into a constrained quadratic programming problem for reducing computational complexity which can be solved by the augmented Lagrange method. The proposed method can significantly improve the performance especially in the scenarios with low signal to noise ratio (SNR), small number of snapshots, and closely spaced correlated sources. In addition, the Cramér-Rao bound (CRB) of the proposed method is developed and the performance guarantee is given according to a version of the restricted isometry property (RIP). The effectiveness and satisfactory performance of the proposed method are illustrated by simulation results