preference of the atoms $\lbra{\m{a}\sbra{f}}$. To be specific, an atom $\m{a}\sbra{f_0}$, $f_0\in\bT$, is more likely selected if $w\sbra{f_0}$ is larger. Moreover, the atomic norm is a special case of the weighted atomic norm with a constant weighting function (i.e., without any preference). Similar to the atomic norm, the proposed weighted atomic norm also admits a semidefinite formulation when assigned an appropriate weighting function, which is stated in the following theorem.

Suppose that $w\sbra{f}= \frac{1}{\sqrt{\m{a}\sbra{f}^H\m{W}\m{a}\sbra{f}}}$ with $\m{W}\in\bC^{N\times N}$. Then, \[thm:weightAN\]

See Appendix \[sec:Append\_weightAN\].

Let $\m{W}_j=\frac{1}{N}\sbra{T\sbra{\m{u}_j}+\epsilon\m{I}}^{-1}$ and $w_j\sbra{f} = \frac{1}{\sqrt{\m{a}\sbra{f}^H \m{W}_j \m{a}\sbra{f}}}$. It follows from Theorem \[thm:weightAN\] that the optimization problem in (\[formu:problem\_j\]) can be exactly written as the following weighted atomic norm minimization problem: As a result, the proposed iterative algorithm can be interpreted as reweighted atomic-norm minimization (RAM), where the weighting function is updated based on the latest solution of $\m{u}$. If we let $w_0(f)$ be a constant function or equivalently, $\m{u}_0=\m{0}$, such that no preference of the atoms is specified at the first iteration, then the first iteration coincides with ANM. From the second iteration on, the preference is defined by the weighting function $w_j\sbra{f}$ given above. Note that $w_j^2(f)$ is in fact the power spectrum of Capon’s beamforming 