Maximum Likelihood Estimation Based Robust Chinese Remainder Theorem for Real Numbers and Its Fast Algorithm

Abstract

Robust Chinese remainder theorem (CRT) has been recently investigated for both integers and real numbers, where the folding integers are accurately recovered from erroneous remainders. In this paper, we consider the CRT problem for real numbers with noisy remainders that follow wrapped Gaussian distributions. We propose the maximum-likelihood estimation (MLE) based CRT when the remainder noises may not necessarily have the same variances. Furthermore, we present a fast algorithm for the MLE based CRT algorithm that only needs to search for the solution among $L$ elements, where $L$ is the number of remainders. Then, a necessary and sufficient condition on the remainder errors for the MLE CRT to be robust is obtained, which is weaker than the existing result. Finally, we compare the performances of the newly proposed algorithm and the existing algorithm in terms of both theoretical analysis and numerical simulations. The results demonstrate that the proposed algorithm not only has a better performance especially when the remainders have different error levels/variances, but also has a much lower computational complexity.