Multi-Dimensional Unlimited Sampling and Robust Reconstruction

TL;DR

This paper introduces a multi-dimensional sampling and reconstruction method based on a new modulo-hysteresis operator, providing robustness and theoretical guarantees for noiseless and noisy signals, extending the Unlimited Sensing Framework.

Contribution

It presents a novel multi-dimensional modulo-hysteresis operator compatible with USF, exploiting higher-dimensional redundancy for robust signal recovery under noise.

Findings

The operator is well-defined with bounded outputs.

Guaranteed input reconstruction in noiseless scenarios.

Error probability decreases with higher sampling rates in noisy conditions.

Abstract

In this paper we introduce a new sampling and reconstruction approach for multi-dimensional analog signals. Building on top of the Unlimited Sensing Framework (USF), we present a new folded sampling operator called the multi-dimensional modulo-hysteresis that is also backwards compatible with the existing one-dimensional modulo operator. Unlike previous approaches, the proposed model is specifically tailored to multi-dimensional signals. In particular, the model uses certain redundancy in dimensions 2 and above, which is exploited for input recovery with robustness. We prove that the new operator is well-defined and its outputs have a bounded dynamic range. For the noiseless case, we derive a theoretically guaranteed input reconstruction approach. When the input is corrupted by Gaussian noise, we exploit redundancy in higher dimensions to provide a bound on the error probability and show this drops to 0 for high enough sampling rates leading to new theoretical guarantees for the noisy case. Our numerical examples corroborate the theoretical results and show that the proposed approach can handle a significantly larger amount of noise compared to USF.