The problem of robustly reconstructing an integer vector from its erroneous remainders appears in many applications in the field of multidimensional (MD) signal processing. To address this problem, a robust MD Chinese remainder theorem (CRT) was recently proposed for a special class of moduli, where the remaining integer matrices left-divided by a greatest common left divisor (gcld) of all the moduli are pairwise commutative and coprime. The strict constraint on the moduli limits the usefulness of the robust MD-CRT in practice. In this paper, we investigate the robust MD-CRT for a general set of moduli. We first introduce a necessary and sufficient condition on the difference between paired remainder errors, followed by a simple sufficient condition on the remainder error bound, for the robust MD-CRT for general moduli, where the conditions are associated with (the minimum distances of) these lattices generated by gcld's of paired moduli, and a closed-form reconstruction algorithm is presented. We then generalize the above results of the robust MD-CRT from integer vectors/matrices to real ones. Finally, we validate the robust MD-CRT for general moduli by employing numerical simulations, and apply it to MD sinusoidal frequency estimation based on multiple sub-Nyquist samplers.

翻译：在多维信号处理领域，从含误差余数中鲁棒重构整数向量的问题具有广泛的应用背景。针对该问题，近年来提出了一类特定模数下的鲁棒多维中国剩余定理（MD-CRT），其要求所有模数通过最大左公因子（gcld）相除后得到的剩余整数矩阵两两互质且可交换。该严格约束限制了鲁棒MD-CRT在实际中的应用。本文针对一般模数集合研究鲁棒MD-CRT。首先，我们引入配对余数误差差的充要条件，并给出余数误差界的简单充分条件，建立了一般模数下鲁棒MD-CRT的理论框架，其中这些条件与配对模数gcld生成的格（的最小距离）相关联，同时提出了一种闭式重构算法。随后，我们将上述鲁棒MD-CRT结果从整数向量/矩阵推广至实数情形。最后，通过数值仿真验证了一般模数下鲁棒MD-CRT的有效性，并将其应用于基于多子奈奎斯特采样器的多维正弦频率估计。