Super-Resolution of Complex Exponentials from Modulations with Unknown Waveforms

TL;DR

This paper introduces a novel method for super-resolution of complex exponentials with unknown waveforms, using atomic norm minimization and low-rank matrix recovery, applicable to applications like imaging and radar.

Contribution

It formulates a non-stationary blind super-resolution problem as a structured low-rank recovery, providing a polynomial-time solution with theoretical guarantees.

Findings

Exact recovery is possible under certain conditions.

The method works with random waveform subspaces.

Numerical simulations validate the approach.

Abstract

Super-resolution is generally referred to as the task of recovering fine details from coarse information. Motivated by applications such as single-molecule imaging, radar imaging, etc., we consider parameter estimation of complex exponentials from their modulations with unknown waveforms, allowing for non-stationary blind super-resolution. This problem, however, is ill-posed since both the parameters associated with the complex exponentials and the modulating waveforms are unknown. To alleviate this, we assume that the unknown waveforms live in a common low-dimensional subspace. Using a lifting trick, we recast the blind super-resolution problem as a structured low-rank matrix recovery problem. Atomic norm minimization is then used to enforce the structured low-rankness, and is reformulated as a semidefinite program that is solvable in polynomial time. We show that, up to scaling ambiguities, exact recovery of both of the complex exponential parameters and the unknown waveforms is possible when the waveform subspace is random and the number of measurements is proportional to the number of degrees of freedom in the problem. Numerical simulations support our theoretical findings, showing that non-stationary blind super-resolution using atomic norm minimization is possible.