剩余数制的鲁棒性
Towards Robustness in Residue Number Systems
摘要：
从其关于若干模的错误解算器中鲁棒地重建大量的问题，即鲁棒解算问题，可能发生在许多应用中，包括相位展开、从若干欠采样波形中检测频率、无线传感器网络等。最近提出了一种鲁棒中国剩余定理（CRT）方法来解决鲁棒排序问题。本文通过放宽动态范围为所有模的lcm的假设，研究了双模系统的动态范围与鲁棒界之间的折衷。它基本上是说，动态范围的减小可能导致鲁棒性界限的增加。我们首先得到一个一般条件的剩余误差，并推导出精确的动态范围与一个封闭形式的公式的鲁棒性举行。然后，我们提出了简单的封闭形式的重建算法。此外，新获得的双模结果应用于多模块系统的鲁棒重构，并推广到实数。最后，进行了一些模拟来验证我们提出的理论结果。
英文摘要：
The problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli, namely the robust remaindering problem, may occur in many applications including phase unwrapping, frequency detection from several undersampled waveforms, wireless sensor networks, etc. Assuming that the dynamic range of the large number is the maximal possible one, i.e., the least common multiple (lcm) of all the moduli, a method called robust Chinese remainder theorem (CRT) for solving the robust remaindering problem has been recently proposed. In this paper, by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli, a trade-off between the dynamic range and the robustness bound for two-modular systems is studied. It basically says that a decrease in the dynamic range may lead to an increase of the robustness bound. We first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed-form formula for the robustness to hold. We then propose simple closed-form reconstruction algorithms. Furthermore, the newly obtained two-modular results are applied to the robust reconstruction for multi-modular systems and generalized to real numbers. Finally, some simulations are carried out to verify our proposed theoretical results.
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