A generalized Chinese remainder theorem (CRT) for multiple integers fromresidue sets has been studied recently, where the correspondence between theremainders and the integers in each residue set modulo several moduli is notknown. A robust CRT has also been proposed lately for robustly reconstruct asingle integer from its erroneous remainders. In this paper, we consider thereconstruction problem of two integers from their residue sets, where theremainders are not only out of order but also may have errors. We prove thattwo integers can be robustly reconstructed if their remainder errors are lessthan $M/8$, where $M$ is the greatest common divisor (gcd) of all the moduli.We also propose an efficient reconstruction algorithm. Finally, we present somesimulations to verify the efficiency of the proposed algorithm. The study ismotivated and has applications in the determination of multiple frequenciesfrom multiple undersampled waveforms.
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机译：最近已经研究了残差集合中多个整数的广义中文剩余定理（CRT），其中余数与每个残差集合中的整数模之间的对应关系是未知的。最近还提出了一种鲁棒的CRT，用于从其错误的余数中鲁棒地重建单个整数。在本文中，我们考虑了两个整数从其残差集的构造问题，其中余数不仅乱序，而且可能存在误差。我们证明了两个整数的余数误差小于$ M / 8 $可以稳健地重构，其中$ M $是所有模数的最大公因数（gcd）。我们还提出了一种有效的重构算法。最后，我们提供了一些仿真来验证所提出算法的效率。该研究是有动机的，并且在从多个欠采样波形确定多个频率中具有应用。
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