A Theory of Computational Resolution Limit for Line Spectral Estimation

• Published in: IEEE transactions on information theory Vol. 67; no. 7; pp. 4812 - 4827
• Main Authors: Liu, Ping, Zhang, Hai
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Estimation; Extraterrestrial measurements; Line spectra; Line spectral estimation; Minimization; Noise level; Phase measurement; phase transition; Phase transitions; Pollution measurement; Random noise; Recovery; resolution limit; Separation; Signal processing; Signal to noise ratio
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2021.3075149

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.