In this paper, we study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements in a bounded domain. We aim to provide a mathematical foundation for sparsity-based super-resolution in such spectral estimation problems in both one- and multi-dimensional spaces. In particular, we estimate the resolution and stability of the location recovery of a cluster of closely spaced point sources when considering the sparsest solution under the measurement constraint, and characterize their dependence on the cut-off frequency, the noise level, the sparsity of point sources, and the incoherence of the amplitude vectors of point sources. Our estimate emphasizes the importance of the high incoherence of amplitude vectors in enhancing the resolution of multi-snapshot spectral estimation. Moreover, to the best of our knowledge, it also provides the first stability result in the super-resolution regime for the well-known sparse MMV problem in DOA estimation.

中文翻译：

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基于稀疏性的多快照谱估计的数学基础


在本文中，我们研究了光谱估计问题，即在有界域中给定傅里叶测量的多个快照来估计固定数量的点源的位置。我们的目标是为一维和多维空间中的此类谱估计问题中基于稀疏性的超分辨率提供数学基础。特别是，我们在测量约束下考虑最稀疏解时估计了一组紧密间隔的点源的位置恢复的分辨率和稳定性，并表征了它们对截止频率、噪声水平、点稀疏性的依赖关系。源，以及点源的幅度矢量的不相干性。我们的估计强调了幅度向量的高度不相干性对于提高多快照谱估计的分辨率的重要性。此外，据我们所知，它还为 DOA 估计中众所周知的稀疏 MMV 问题提供了超分辨率机制中的第一个稳定性结果。