On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia and his collaborators proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are dm1, dm2, · · · , dmk with m1,m2, · · · ,mk being pairwise coprime. If the errors of the corrupted remainders are within d 4 , their schemes are able to construct an approximation of the solution to the generalized CRT with an error smaller than d 4 . One of the critical ingredients in their approach is the clever idea of accurately finding the quotients. In this paper, we present two treatments of this problem. The first treatment follows the route of Wang and Xia to find the quotients, but with a simplified process. The second treatment takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo d. Both of our treatments produce efficient algorithms with essentially optimal performance. This paper also provides a proof of the sharpness of the error bound d 4 .