# Blind Unwrapping of Modulo Reduced Gaussian Vectors: Recovering MSBs from LSBs

## TL;DR

This paper introduces a low-complexity iterative decoding algorithm for recovering Gaussian samples from modulo reduced measurements, closely matching the performance of an ideal decoder with known covariance, even in finite sample scenarios.

## Contribution

The paper proposes a novel blind decoding algorithm for Gaussian vectors from modulo measurements, achieving near-optimal performance without prior covariance knowledge.

## Key findings

- Algorithm performs well in non-asymptotic conditions
- Performance closely matches informed decoders with known covariance
- Effective for quantization and analog-to-digital conversion scenarios

## Abstract

We consider the problem of recovering $n$ i.i.d samples from a zero mean multivariate Gaussian distribution with an unknown covariance matrix, from their modulo wrapped measurements, i.e., measurement where each coordinate is reduced modulo $\Delta$, for some $\Delta>0$. For this setup, which is motivated by quantization and analog-to-digital conversion, we develop a low-complexity iterative decoding algorithm. We show that if a benchmark informed decoder that knows the covariance matrix can recover each sample with small error probability, and $n$ is large enough, the performance of the proposed blind recovery algorithm closely follows that of the informed one. We complement the analysis with numeric results that show that the algorithm performs well even in non-asymptotic conditions.