Spectral Super-resolution With Prior Knowledge

TL;DR

This paper introduces a semidefinite programming approach for super-resolution frequency recovery that leverages prior knowledge about the signal spectrum, enabling perfect reconstruction with fewer samples and improved resolution.

Contribution

It presents a novel method combining prior spectral information with positive trigonometric polynomial theory for enhanced super-resolution frequency recovery.

Findings

Perfect reconstruction possible with samples up to three times the number of frequencies.

Method significantly improves resolution and performance over existing techniques.

Numerical experiments validate the effectiveness of incorporating prior knowledge.

Abstract

We address the problem of super-resolution frequency recovery using prior knowledge of the structure of a spectrally sparse, undersampled signal. In many applications of interest, some structure information about the signal spectrum is often known. The prior information might be simply knowing precisely some signal frequencies or the likelihood of a particular frequency component in the signal. We devise a general semidefinite program to recover these frequencies using theories of positive trigonometric polynomials. Our theoretical analysis shows that, given sufficient prior information, perfect signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies. Numerical experiments demonstrate great performance enhancements using our method. We show that the nominal resolution necessary for the grid-free results can be improved if prior information is suitably employed.