# Simultaneous Sparse Recovery and Blind Demodulation

## TL;DR

This paper introduces a novel method for simultaneous sparse signal recovery and blind demodulation, leveraging atomic norm minimization, with theoretical guarantees and practical validation in noisy and noiseless scenarios.

## Contribution

It proposes a new approach using lifting and atomic norm minimization for joint sparse recovery and blind demodulation with theoretical sample complexity bounds.

## Key findings

- Near optimal sample complexity bounds for perfect recovery.
- Effective recovery in noisy conditions demonstrated.
- Theoretical analysis supported by numerical simulations.

## Abstract

The task of finding a sparse signal decomposition in an overcomplete dictionary is made more complicated when the signal undergoes an unknown modulation (or convolution in the complementary Fourier domain). Such simultaneous sparse recovery and blind demodulation problems appear in many applications including medical imaging, super resolution, self-calibration, etc. In this paper, we consider a more general sparse recovery and blind demodulation problem in which each atom comprising the signal undergoes a distinct modulation process. Under the assumption that the modulating waveforms live in a known common subspace, we employ the lifting technique and recast this problem as the recovery of a column-wise sparse matrix from structured linear measurements. In this framework, we accomplish sparse recovery and blind demodulation simultaneously by minimizing the induced atomic norm, which in this problem corresponds to the block $\ell_1$ norm minimization. For perfect recovery in the noiseless case, we derive near optimal sample complexity bounds for Gaussian and random Fourier overcomplete dictionaries. We also provide bounds on recovering the column-wise sparse matrix in the noisy case. Numerical simulations illustrate and support our theoretical results.