# Identifiability Conditions for Multi-channel Blind Deconvolution with Short Filters

## TL;DR

This paper establishes conditions for the well-posedness of multi-channel blind deconvolution with short filters and demonstrates an effective nuclear norm minimization approach through numerical experiments.

## Contribution

It derives necessary and sufficient conditions for identifiability and reformulates the problem as a low-rank matrix recovery task solved by nuclear norm minimization.

## Key findings

- Effective algorithm under certain generative models
- Successful in noiseless and noisy scenarios
- Provides theoretical identifiability conditions

## Abstract

This work considers the multi-channel blind deconvolution problem under the assumption that the channels are short. First, we investigate the ill-posedness issues inherent to blind deconvolution problems and sufficient and necessary conditions on the channels that guarantee well-posedness are derived. Following previous work on blind deconvolution, the problem is then reformulated as a low-rank matrix recovery problem and solved by nuclear norm minimization. Numerical experiments show the effectiveness of this algorithm under a certain generative model for the input signal and the channels, both in the noiseless and in the noisy case.