Compressed sensing consists in taking non-adaptive randomized samples of a sparse signal, which can then be estimated robustly by solving an optimization problem. However, in applications the sensing mechanism may be constrained by practical considerations. In such cases, optimization-based methods are often still very useful, but to characterize their performance we must first analyze the interaction between the measurement operator and the signals of interest. We illustrate this by considering the problem of estimating point sources from bandlimited data. We argue that conditions precluding the sources from being too clustered are necessary for the problem not to be hopelessly ill posed. Then, we show that under such a condition solving a tractable convex program allows to accurately estimate the sources even in the presence of noise.
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