Optimal Estimates of Two Common Remainders for a Robust Generalized Chinese Remainder Theorem

Estimation of multiple common remainders from a sequence of erroneous residue sets is an important step for the robust generalized Chinese Remainder Theorem (CRT). This paper considers the problem of how to estimate the two common remainders from their residue sets modulo a set of moduli. To measure the errors properly under the modular operation, we introduce two type circular distances. Based on these circular distances, two estimation methods are proposed by properly grouping the erroneous remainders into two ordered clusters. Both of the two methods perform better with lower computational complexities than the existing methods. In this paper, theoretical analysis as well as analytical results for the two proposed methods are obtained. For the first method, the two optimal estimates are proved to be in a finite set with no more than $L$ candidates, where $L$ is the number of the given moduli. The second method have closed forms. Simulation results show that the two proposed methods have nearly the same performance. These optimal estimates can improve the performance of the robust generalized CRT significantly.