Optimal estimates of common remainder for the robust Chinese Remainder Theorem

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Highlights

  • We give two optimal estimates of common remainder for the robust Chinese Remainder Theorem.

  • Two different optimal estimates are obtained based on the different definitions of circular distance.

  • Simulation results show that the two optimal estimates have nearly the same performance.

  • The second optimal estimate has less computational complexity than the first one.

  • These optimal estimates can improve the efficiency of estimating the common remainder.

Abstract

Common remainder is significant to the estimation of the robust Chinese Remainder Theorem (CRT). This paper presents two optimal estimates of common remainder for the robust CRT. The two different optimal estimates are obtained based on different definitions of circular distance. Both of the two estimations are more effective with lower computational complexity than the existing searching method. Simulation results show that the two estimations have nearly the same performance, however, the second optimal estimation has less computation than the first one. These optimal estimates can improve the performance of the estimation of the robust CRT.

Introduction

Chinese Remainder Theorem (CRT) is an old research topic, which tells that a positive large integer can be reconstructed from its remainders [1], [2]. It has a simple formula to determine the original integer from the remainder sequence when all of the moduli are co-prime. When the moduli are not co-prime, the reconstruction is unique if and only if the integer is less than the least common multiple (lcm) of all the moduli. However, CRT is well-known not robust, i.e., a small error in any remainder may cause a large reconstruction error to the large integer. In practical applications, the remainders usually have errors because of noises, which often appear in engineering problems (see for example in [3], [4], [5], [6]).

In order to resist remainder errors, two kinds of methods are proposed in the literature, i.e., remainder number redundancy methods [7], [8], [9], [10] and remainder redundancy methods [12], [13], [11], [14], [15]. The first kind of methods are based on the fact that sufficient moduli are used so that only a few of the remainders with errors does not affect the reconstruction accuracy. For example, in [8], the integer is correctly recovered from its residues modulo sufficient primes, when the residues are corrupted by a small additive noise bounded in the Lee norm. In the second methods, all the moduli have a common greatest common divisor (gcd) larger than 1, and all the remainders are allowed to have errors which may not be too large, then the integer can be robustly reconstructed. The second kind of methods, called the robust CRT, have numerous applications in physical and engineering problems, such as frequency estimation from under sampled waveforms [12], [13], radar signal processing [14], [15], [16], [17], distance estimation [18], [19], etc.

We will focus on the second kind of error resisting of remainders in CRT, that is, we will consider the case that all the moduli have the same factor which is larger than 1. From [17], [20] we note that the estimation of common remainder is the key step for the robust estimating of the number. Motivated by this conclusion, we present the optimal estimates of the common remainder in this paper. In order to obtain the optimal common remainder, we introduce two kinds of circular distance which are different from [21], [22], [23]. The optimal estimate is then obtained by each definition. The simulation results show that these estimations have nearly the same performance.

The paper is organized as follows. Section 2 introduces two optimal estimates of the common remainder for the robust CRT. In Section 3, we demonstrate the results of Monte Carlo experiment. We end with concluding remarks in Section 4.

Section snippets

The extended CRT and estimation of the common remainder

In this section, we first recall the basic idea of the extended CRT and then introduce the issue of the robust estimation of the common remainder from contaminated data.

Let N be a positive integer, 0<M1<M2<<ML be the L moduli, and r1,r2,,rL be the L remainders of N. The extended CRT is the special case of traditional CRT when the gcd of all the moduli is M(M1) and the remaining integers factorized by the gcd are co-prime. Assume that Mi=MΓi,i=1,2,,L, and Γ1,Γ2,,ΓL are all co-prime

Monte Carlo Simulations

In this section we present some simulations to verify the efficiency of the two optimal estimations which were proposed in this paper. In the simulations, M=100,L=6, and Γ1 to Γ6 are 3,5,7,11,13,17, respectively. N is a real number and uniformly distributed between 0 to 25525500, and the number of the simulations is 10000. The remainders’ errors are independent and identical normal distribution random variable with zero mean and σ2 variance. We introduce the Signal to Noise Ratio (SNR) to

Conclusions

In this paper, based on two definitions of the circular distance, we have presented two optimal estimations of the common remainder for the robust CRT. From the previous theoretical analysis, we observe that for estimating the common remainder, either the first or the second optimal estimation is much faster than the existing searching methods. Simulation results show the efficiency of the two methods. Besides, we can see from the simulations that two optimal estimations have nearly the same

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    This work was supported by the National Natural Science Foundation of China (Grants Nos. 61172092, 61302069) and the Research Fund for the Doctoral Programs of Higher Education of China (No. 20130201110014).

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