Robust Multidimentional Chinese Remainder Theorem for Integer Vector Reconstruction

Abstract

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.