The Residue Number System With Gaussian Noise

Residue number system (RNS) is a concept of number theory which represents an integer as a set of its residues or remainders and usually utilizes Chinese Remainder Theorem (CRT) to reconstruct the integer from its multiple residues. We consider the phase unwrapping problem by using RNS in this paper. We consider the RNS which all of the moduli have the same factor M in order to resist the noise sensitivity. A robust Chinese Remainder Theorem was introduced and generalized to real domain in order to resolve that a small error in a remainder may cause a large error in the reconstruction. We obtain some useful conclusions about the robust CRT and propose a new robust CRT algorithm at the same time. We simulated the algorithm when the number of the remainders L is8by using the Mento Carlo method. The distance estimation which proposed in the section2.1was solved by using the robust CRT method when SNR is16dB. The results tell us that the algorithm can overcome the noisy sensitivity. At last, we propose a maximum likelihood estimation (MLE) algorithm and new fast algorithm in the viewpoint of detection and estimation. We simulated the algorithm when the number of the remainders L is3. Compared with the MLE algorithm, new algorithm have more higher precision and faster operation. At last, the performance of the algorithm are given out. It is show that the estimation of the common remainder is significant. We have proved that the optimal estimation of the common remainder is one of the element of the set which has L elements. Thus, the number of searching is reduced sharply. The sufficiency and necessity condition is obtained with the common remainder consequently.