# Atomic Norm Denoising for Complex Exponentials with Unknown Waveform Modulations

## TL;DR

This paper introduces an atomic norm regularized approach for denoising signals composed of complex exponentials with unknown modulations in non-stationary blind super-resolution, providing theoretical error bounds and numerical validation.

## Contribution

It extends atomic norm methods to denoise complex exponential signals with unknown waveforms, offering new theoretical insights and practical algorithms.

## Key findings

- Mean square error depends on noise variance and signal parameters.
- The proposed method achieves robust denoising in non-stationary blind super-resolution.
- Numerical experiments validate the theoretical error bounds.

## Abstract

Non-stationary blind super-resolution is an extension of the traditional super-resolution problem, which deals with the problem of recovering fine details from coarse measurements. The non-stationary blind super-resolution problem appears in many applications including radar imaging, 3D single-molecule microscopy, computational photography, etc. There is a growing interest in solving non-stationary blind super-resolution task with convex methods due to their robustness to noise and strong theoretical guarantees. Motivated by the recent work on atomic norm minimization in blind inverse problems, we focus here on the signal denoising problem in non-stationary blind super-resolution. In particular, we use an atomic norm regularized least-squares problem to denoise a sum of complex exponentials with unknown waveform modulations. We quantify how the mean square error depends on the noise variance and the true signal parameters. Numerical experiments are also implemented to illustrate the theoretical result.