Resolution Limit for Line Spectral Estimation: Theory and Algorithm
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Line spectral estimation is a classical signal processing problem aimed to estimate the spectral lines of a signal from its noisy (deterministic or random) measurements. Despite a large body of research on this subject, the theoretical understanding of the spectral line estimation is still elusive. In this paper, we quantitatively characterize the two resolution limits in the line spectral estimation problem: one is the minimum separation distance between the spectral lines that is required for an exact recovery of the number of spectral lines, and the other is the minimum separation distance between the spectral lines that is required for a stable recovery of the supports of the spectral lines. The quantitative characterization implies a phase transition phenomenon in each of the two recovery problems, and also the subtle difference between the two. Moreover, they give a sharp characterization to the resolution limit for the deconvolution problem as a consequence. Finally, we proposed a recursive MUSIC-type algorithm for the number recovery and an augmented MUSIC-algorithm for the support recovery, and analyze their performance both theoretically and numerically. The numerical results also confirm our results on the resolution limit and the phase transition phenomenon.
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