  In estimating frequencies given that the signal waveforms are undersampled
multiple times, Xia et. al. proposed to use a generalized version of Chinese
remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are
not necessarily pairwise coprime. If the errors of the corrupted remainders are
within $\tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}}
\frac{\gcd(M_i,M_j)}4$, their schemes can be used to construct an approximation
of the solution to the generalized CRT with an error smaller than $\tau$.
Accurately finding the quotients is a critical ingredient in their approach. In
this paper, we shall start with a faithful historical account of the
generalized CRT. We then present two treatments of the problem of solving
generalized CRT with erroneous remainders. The first treatment follows the
route of Wang and Xia to find the quotients, but with a simplified process. The
second treatment considers a simplified model of generalized CRT and 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 $4\tau$. Both of
our treatments produce efficient algorithms with essentially optimal
performance. Finally, this paper constructs a counterexample to prove the
sharpness of the error bound $\tau$.
