A Robust Generalized Chinese Remainder Theorem for Two Integers

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 to 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 not only are 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 of all the moduli. We also propose an efficient reconstruction algorithm. Finally, we present some simulations to verify the efficiency of the proposed algorithm. This paper is motivated from and has applications in the determination of multiple frequencies from multiple undersampled waveforms.