Single-Snapshot Nested Virtual Array Completion: Necessary and Sufficient Conditions

We study the problem of completing the virtual array of a nested array with a single snapshot. This involves synthesizing a virtual uniform linear array (ULA) with the same aperture as the nested array by estimating (or interpolating) the missing measurements. A popular approach for virtual array synthesis involves completing a certain Hankel/Toeplitz matrix from partial observations, by seeking low-rank solutions. However, existing theoretical guarantees for such structured rank minimization (which mostly provide sufficient conditions) do not readily extend to nested arrays. We provide the first necessary and sufficient conditions under which it is possible to exactly complete the virtual array of a nested array by minimizing the rank of a certain Toeplitz matrix constructed using a single temporal snapshot. Our results exploit the geometry of nested arrays and do not depend on the source configuration or on the separation between sources.