Communications in Nonlinear Science and Numerical Simulation
Optimal estimates of common remainder for the robust Chinese Remainder Theorem☆
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, be the L moduli, and be the L remainders of N. The extended CRT is the special case of traditional CRT when the gcd of all the moduli is and the remaining integers factorized by the gcd are co-prime. Assume that , and 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, , and to are , 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 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
References (23)
- et al.
Noise and coupling induced synchronization in a network of chaotic neurons
Commun Nonlinear Sci Numer Simul
(2013) Reliable amplitude and frequency estimation for biased and noisy signals
Commun Nonlinear Sci Numer Simul
(2011)- et al.
Denoising signals corrupted by chaotic noise
Commun Nonlinear Sci Numer Simul
(2010) - et al.
Noisy Chinese remaindering in the Lee norm
J Complexity
(2004) - et al.
Chinese remainder theorem: applications in computing, coding, cryptography
(1999) Elementary number theory and its applications
(2010)- et al.
Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh–Nagumo neural model
Commun Nonlinear Sci Numer Simul
(2013) - et al.
Chinese remaindering with errors
IEEE Trans Inf Theory
(2000) - et al.
A coding theory approach to error control in redundant residue number systems-part II: multiple error detection and correction
IEEE Trans Circuits Syst
(1992) - et al.
Systematic redundant residue number system codes: analytical upper bound and iterative decoding performance over AWGN and rayleigh channels
IEEE Trans Comput
(2006)
Multiple error detection and correction based on redundant residue number systems
IEEE Trans Commun
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