Line spectral estimation is a classical signal processing problem that aims to estimate the line spectra from their signal which is contaminated by deterministic or random noise. Despite a large body of research on this subject, the theoretical understanding of this problem is still elusive. In this paper, we introduce and quantitatively characterize the two resolution limits for the line spectral estimation problem under deterministic noise: one is the minimum separation distance between the line spectra that is required for exact detection of their number, and the other is the minimum separation distance between the line spectra that is required for a stable recovery of their supports. The quantitative results imply a phase transition phenomenon in each of the two recovery problems, and also the subtle difference between the two. We further propose a sweeping singular-value-thresholding algorithm for the number detection problem and conduct numerical experiments. The numerical results confirm the phase transition phenomenon in the number detection problem.

翻译：光谱估计是一个典型的信号处理问题,目的是从其受确定性或随机噪音污染的信号中估计线光谱。尽管对这个问题进行了大量研究,但对这一问题的理论理解仍然难以实现。在本文中,我们引入了确定性噪音下线光谱估计问题的两个分辨率极限,并定量地描述这两个分辨率限度:一个是线光谱之间的最小分离距离,这是准确检测其数量所需要的,另一个是线光谱之间的最小分离距离,这是稳定恢复其支持所需的。量化结果意味着两个恢复问题中的每一个问题都存在一个阶段过渡现象,以及两者之间的微妙差异。我们进一步建议对数字探测问题和进行数字实验采用一个全面的单值持有算法。数字结果证实了数字探测问题中的阶段过渡现象。