Towards Robustness in Residue Number Systems

TL;DR

This paper explores the trade-off between dynamic range and robustness in residue number systems, proposing new algorithms and conditions for improved error correction in multi-modular reconstructions.

Contribution

It relaxes the fixed dynamic range assumption, deriving a closed-form formula for robustness and proposing simple reconstruction algorithms for two-modular systems.

Findings

Decreased dynamic range can increase robustness bounds.

Derived exact dynamic range with a closed-form formula.

Validated results through simulations.

Abstract

The problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli, namely the robust remaindering problem, may occur in many applications including phase unwrapping, frequency detection from several undersampled waveforms, wireless sensor networks, etc. Assuming that the dynamic range of the large number is the maximal possible one, i.e., the least common multiple (lcm) of all the moduli, a method called robust Chinese remainder theorem (CRT) for solving the robust remaindering problem has been recently proposed. In this paper, by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli, a trade-off between the dynamic range and the robustness bound for two-modular systems is studied. It basically says that a decrease in the dynamic range may lead to an increase of the robustness bound. We first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed-form formula for the robustness to hold. We then propose simple closed-form reconstruction algorithms. Furthermore, the newly obtained two-modular results are applied to the robust reconstruction for multi-modular systems and generalized to real numbers. Finally, some simulations are carried out to verify our proposed theoretical results.