$\norm{\m{Y}}_{\cA,0}$ and is expected to enhance sparsity and resolution compared to $\atomn{\m{Y}}$. Moreover, Theorem \[thm:epsilontozero\] characterizes the properties of the optimizer $\m{u}_{\epsilon}^*$ as $\epsilon\rightarrow0$ including the convergent speed of the smallest $N-\norm{\m{Y}}_{\cA,0}$ eigenvalues and the limiting form of $T\sbra{\m{u}^*}$ via the Vandermonde decomposition. In fact, we always observe via simulations that the smallest $N-\norm{\m{Y}}_{\cA,0}$ eigenvalues of $T\sbra{\m{u}^*}$ become zero once $\epsilon$ is appropriately small.

In DOA estimation, a difficult scenario is when the source signals $\lbra{\m{s}_k}_{k=1}^K$ are highly or even completely correlated (the latter case is usually called coherent). For example, covariance-based methods like Capon’s beamforming and MUSIC cannot produce satisfactory results since a faithful covariance estimate is unavailable. In contrast, the proposed sparse metric is robust to source correlations by Theorem \[thm:epsilontozero\] in which we have not made any assumption for the sources.

According to Theorem \[thm:epsilontozero\], the solution $T\sbra{\m{u}^*}$ can be interpreted as the data covariance of $\m{Y}$ after removing correlations among the sources. \[rem:datacov\]

Reweighted Atomic-Norm Minimization (RAM) {#sec:RAM}
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A Locally Convergent Iterative Algorithm
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Using the proposed sparse metric $\cM^{\epsilon}\sbra{\m{Y}}$ we solve the following optimization problem for signal and frequency recovery: or equivalently, Note that $\ln\abs{T\sbra{\m{u}}+\epsilon\m{I}}$ is a concave function of $\m{u}$ since $\ln\abs{\m{R}}$ is 