A Robust Generalized Chinese Remainder Theorem for Two Integers

TL;DR

This paper introduces a robust generalized Chinese Remainder Theorem for reconstructing two integers from erroneous and unordered residue sets, with proven error bounds and an efficient algorithm, applicable in frequency determination from undersampled signals.

Contribution

It extends the CRT to handle two integers with errors and unordered residues, providing a new error bound and an efficient reconstruction algorithm.

Findings

Robust reconstruction is possible if errors are less than M/8.

The proposed algorithm is efficient and verified through simulations.

Applications include frequency determination from undersampled waveforms.

Abstract

A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known. A robust CRT has also been proposed lately for robustly reconstruct a single integer from its erroneous remainders. In this paper, we consider the reconstruction problem of two integers from their residue sets, where the remainders are not only out of order but also may have errors. We prove that two integers can be robustly reconstructed if their remainder errors are less than $M/8$, where $M$ is the greatest common divisor (gcd) of all the moduli. We also propose an efficient reconstruction algorithm. Finally, we present some simulations to verify the efficiency of the proposed algorithm. The study is motivated and has applications in the determination of multiple frequencies from multiple undersampled waveforms.