Statistical Robust Chinese Remainder Theorem for Multiple Numbers

Guoqiang Xiao; Hanshen Xiao; Nan Du; Zhikang T. Wang

REVIEW 4 major objections 4 minor 31 references

Statistical Robust Chinese Remainder Theorem for Multiple Numbers

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Statistical robust CRT can be solved by a MAP residue clustering found among O(NL) candidates.

desk verdict A genuinely new statistical framing of GRCRT with promising heuristics, but the advertised MAP derivation is invalid because Assumption 1 is not independent of the permutation K. read the letter →

arxiv 1909.00225 v1 pith:R3MA5ACA submitted 2019-08-31 stat.OT math.CO

Hanshen Xiao , Nan Du , Zhikang T. Wang , Guoqiang Xiao This is my paper · ORCID

classification stat.OTmath.CO MSC 11A0762F1562H3094A12

keywords robustChineseremaindertheoremmultipleparameterestimationmaximumaposterioriwrappedGaussianmixturemodelresidueclusteringambiguityresolutionerrorcorrectingcodesdynamicrange

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the robust reconstruction of $N$ unknown numbers from $L$ noisy, unordered residue sets—the generalized robust Chinese remainder problem—can be solved statistically rather than only deterministically. It proposes a statistical model in which the unknown residue-to-number correspondence is a random permutation and the noisy residues wrap onto a circle of length $\Gamma$. Under an assumption that each number’s noise spreads over less than half that circle, the first algorithm computes the MAP residue clustering from $O(NL)$ candidates, reducing the task to $N$ independent single-number robust CRT problems; the second algorithm iteratively alternates between matching residues to common residues and refining those common residues. If correct, these schemes operate at the full dynamic range $D=\mathrm{lcm}(m_1,\ldots,m_L)$ instead of a shrunken range, and they outperform deterministic schemes under heavy noise, while the paper notes that deterministic methods can win at very low noise.

What carries the argument

The load-bearing object is the common residue $\mu_i=\langle Y_i\rangle_\Gamma$—the residue every observation of $Y_i$ shares on the small circle of length $\Gamma$—together with the cutting point $\tau$ of Assumption 1. Cutting the circle at $\tau$ straightens the noise intervals into ordered segments, so the ranks of the projected residues $r_{il}$ correspond to the ranks of the noise terms. The MAP objective becomes a quadratic form that, by the rearrangement inequality, is maximized by sorting-matching the projected residues across samplers, reducing the search over $L\cdot N!$ permutations to a scan of $NL$ candidate cutting points.

What would settle it

Take $N=2$, $L=2$, $\Gamma=1$, and noise standard deviation $\sigma=0.4$; numerically integrate the true posterior in (5) over all residue clusterings and compare the resulting argmax with Algorithm 1’s ranking of the $O(NL)$ candidate cutting points. If equation (13) were exact, the two rankings would coincide; as $\sigma$ approaches $\Gamma$, ranking mismatches should grow, showing that the closed form is not the true posterior in heavy noise.

Watch

Extended reading notes

Core claim

The central discovery is that the correspondence ambiguity—which noisy residue belongs to which underlying number—can be treated as a parameter to be estimated by maximum a posteriori probability, and that under an assumption on how far noise spreads around a small circle of circumference $\Gamma$, the MAP clustering reduces to an ordering problem. Projecting each observation $R_{il}$ to its residue $r_{il}=\langle R_{il}\rangle_\Gamma$ reveals a common residue $\mu_i=\langle Y_i\rangle_\Gamma$ shared by all $L$ moduli. For a fixed cutting point $\tau$, the optimal clustering groups the $i$-th largest elements of each sampler’s projected residues together, a consequence of the rearrangement inequality, and scanning the $NL$ distinct observed projected residues as cutting points gives the global MAP clustering under the assumption. After clustering, reconstruction decomposes into $N$ independent single-number robust CRT problems. The second algorithm replaces the fixed cutting point by an iteration that alternates between circular matching of residues to current common-residue estimates and least-squares updating of those estimates.

Load-bearing premise

The load-bearing premise is that integrating a product of wrapped Gaussian densities over one period equals integrating the product of unwrapped Gaussians over the whole line; that equality is exact only when the noise is much smaller than the period $\Gamma$, and the paper applies it precisely in the heavy-noise regime where the two sides diverge.

Editorial extensions

If this is right

Under Assumption 1, the MAP residue clustering is computed from $O(NL)$ candidates rather than $L\cdot N!$, so the correspondence bottleneck disappears and each number is reconstructed by a single-number RCRT.
Statistical RCRT can work with dynamic range $D=\mathrm{lcm}(m_1,\ldots,m_L)$, the maximal moduli utilization, whereas deterministic generalized robust CRT requires a shrunken range and a noise bound of $\Gamma/(4N)$.
Algorithm 2’s iterative wrapped-Gaussian mixture refinement gives stronger robustness than Algorithm 1, at the cost of more computation, and in the paper’s simulations it reaches a stationary state within a few iterations, mostly between two and three.
Residue error-correcting codes, used with Theorem 3, tolerate a bounded number of clustering errors; this becomes more beneficial as $N$ grows, when a larger number of moduli can be exploited for correction.
The paper itself states that in the extremely low noise case, deterministic schemes may outperform the proposed schemes, consistent with the theoretical analysis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the exact-MAP claim depends on the wrapped/unwrapped integral identity in Lemma 3; if that identity is not exact in heavy noise, the $O(NL)$ search produces an approximation, though still a tractable clustering rule.
Editorial inference: the sorting-matching step is a circular assignment problem, so optimal transport or assignment algorithms may extend the method to regimes where Assumption 1 fails or to non-Gaussian circular noise.
Editorial inference: a numerical comparison against exact marginalization for small $N$ and $L$ would settle whether Algorithm 1 is truly MAP or only a high-quality heuristic in heavy noise.
Editorial inference: the same “common residue plus circular sorting” mechanism may apply to other one-dimensional circular registration problems, such as multi-sensor phase unwrapping or circular motif detection.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know before you read this one. First, it is a real attempt at something new: a statistical treatment of generalized CRT for multiple numbers with unordered residues, framed as a wrapped Gaussian mixture, and it ships two concrete algorithms with plausible simulations. Second, the central theoretical claim—that Algorithm 1 computes the MAP of residue clustering under Assumption 1—is wrong. The independence argument between Eqs. (11) and (12) fails, so the posterior being maximized is not the conditional MAP.

What is good: the problem setup is well motivated. The paper distinguishes correspondence ambiguity from perturbation, and the connection between the common residue µ_i and the clustering of ⟨Ril⟩_Γ is a useful observation. Algorithm 1's O(NL) enumeration via the rearrangement inequality is neat, given a fixed cut point τ. Algorithm 2, the iterative wrapped-Gaussian EM, is sensible as a heuristic, and the experiments show clear gains over the deterministic baseline in heavy noise. The error-correcting-code addition is a decent touch.

Where it falls: the load-bearing flaw is the claim that Pr(A1|K) is constant across K. The proof asserts that because p(R|K) is Gaussian, the integral with p(A1|K,R) is constant. That does not follow. A concrete counterexample: Γ=1, N=3, L=3, zero noise. Identity permutation puts each cluster as a singleton, so A1 holds with probability 1. A cyclic permutation groups one residue of each Y_i into each cluster; then A1 holds only if the three Y_i lie in some open semicircle, probability 3/4. So P(A1|K) varies, and p(K|R,A1) contains a K-dependent factor that Theorem 1 drops. Algorithm 1 therefore maximizes something close to the likelihood, not the MAP. Eq. (13) is less of a problem than the reader's first pass suggested—under A1 the wrapped/unwrapped likelihood is defensible. The real issue is the independence.

Bottom line: the empirical work may be salvageable as a heuristic, and the problem is worth studying, but as written the paper overclaims. It deserves serious refereeing because the algorithms and motivation are substantive; just not acceptance in this form unless the MAP claim is corrected or the paper is repositioned as a heuristic with empirical support.

Referee Report

4 major / 4 minor

Summary. The paper studies the reconstruction of N real numbers Y_i from unordered noisy residues modulo m_l = Γ M_l, where the M_l are pairwise coprime and D = Γ∏ M_l is the maximal dynamic range. It proposes two statistical schemes: Algorithm 1, which claims to compute the MAP of residue clustering under Assumption 1 in semi-closed form by enumerating O(NL) cut points, followed by single-number RCRT; and Algorithm 2, an iterative wrapped-Gaussian mixture procedure alternating circular nearest-neighbor matching and common-residue estimation. Residue error-correcting codes and majority voting over modulus subsets are added as robustness enhancements. Simulations compare success rates against the deterministic GRCRT of [25] for N=2,...,10 over SNR -40 to 0 dB.

Significance. If the theoretical claims were correct, the paper would be a substantial step: it promises maximal dynamic range D=lcm(m_l) while simultaneously resolving the correspondence ambiguity and tolerating noise, with polynomial-time algorithms. The algorithmic recipes are concrete, the modular-projection viewpoint is sound, and the simulation comparison is extensive. However, the central theoretical foundation is not established: the likelihood identity used to derive the MAP is not exact in the regime of interest, and the global conditioning on Assumption 1 is not handled correctly. The contribution as a rigorous statistical framework therefore fails, although the empirical heuristics may still be of practical interest.

major comments (4)

[Section III, Lemma 3] Lemma 3 replaces the product of wrapped Gaussian likelihoods by a product of real-line Gaussian integrals. This is not an identity for L>1. For L=2 with equal weights, the exact marginal likelihood for two residues r1,r2 assigned to one number is ∫_0^Γ w_σ(r1|μ)w_σ(r2|μ)dμ = (1/Γ)∑_{k∈Z} e^{2πik(r1-r2)/Γ} e^{-4π²σ²k²/Γ²}, a periodic function of r1-r2. Eq. (13) instead gives exp[-(r1-r2)²/(4σ²)] up to constants. When σ is comparable to Γ, the exact likelihood is nearly uniform in the circular difference, whereas the real-line expression still discriminates strongly; for σ=Γ and |r1-r2|=Γ/2 the real-line factor is e^{-1/16}≈0.94, while the wrapped likelihood differs from its maximum by a factor of order e^{-4π²}. Therefore Theorem 1 and Algorithm 1 maximize an approximate objective, not the claimed posterior MAP. The equality in Eq. (7) is special to L=1 and does not extend to the product in Eq. (6).
[Section III, Lemma 3] Eq. (13) also asserts factorization over the N clusters. Assumption 1 is a global event: it requires a single point τ lying outside the union of all intervals I_i, i=1,...,N. The indicator of this event in the numerator p(R,A1|K)=∫ p(R|Y,K) 1_{A1} dY couples all i, so the conditional density does not factorize into per-cluster integrals as the product in Eq. (13) claims. The sentence 'due to the independence of μ_i' does not address the existence of a common τ. Even if one fixes a valid τ, the integration over each μ_i should be restricted to the small arc allowed by the event rather than extended to the whole real line. Thus the semi-closed-form MAP derivation is not established.
[Section III, Eqs. (11)-(12)] The proof that Assumption 1 is independent of the permutation K is invalid as written. The text after Eq. (12) asserts that the integral ∫ p(A1|K,R)p(R|K)dR is constant in K because p(R|K) is a normal distribution. But p(R|K) is a product of wrapped Gaussian densities, not a normal density, and p(A1|K,R) is an indicator that depends on the cluster geometry induced by K. The stated reason does not imply constancy. The conclusion may be salvageable by an exchangeability argument applied to the i.i.d. noise matrix, since K only permutes entries within each column, but the proof given must be replaced if the paper is to claim a rigorous derivation.
[Section IV, Eq. (24) and Section V] Step one of Algorithm 2 replaces the wrapped likelihood in Eq. (23) by the nearest-centroid objective (24), justified only when σ_l² is much smaller than Γ. The simulations in Section V set σ²=10^{-SNR/10}, with Γ=100, so at SNR=-40 dB one has σ=100=Γ. The heavy-noise regime in which the paper claims large performance gains is therefore exactly the regime where the theoretical justification of Eq. (24) fails. The experimental results may reflect a useful heuristic, but they are not backed by the paper's statistical derivation.

minor comments (4)

[General] There are several typos: 'Methology' in Section II, 'stead state' in Section IV, and 'K[1:N]' in Eq. (19) where the permutation variable is indexed by L, not N.
[Algorithm 2] The input line 'Calculate ril = ⟨\tilde r_il⟩_{M_l}' appears to be a typo; the algorithm should compute the residue of R_il modulo Γ, i.e., r_il = ⟨R_il⟩_Γ.
[Section IV, Eq. (29)] The formula in Eq. (29) is garbled: the weighted-average expression should involve the unwrapped values γ_(l)+Γ for l<j, and the displayed index ranges appear to double-count terms. Please correct the expression and its derivation.
[Section III, Theorem 1] The paper states that the MAP can be found from O(NL) candidate cut points, but each candidate requires re-sorting the residue sets; if a total complexity claim is intended, it should be stated explicitly rather than leaving the reader to infer it from the number of candidates.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the central MAP derivation is self-contained and cited prior results are independent supports rather than fitted inputs.

full rationale

The paper's central claim is a Bayesian derivation of residue clustering under a stated generative model: uniform Y_i in [0,D), i.i.d. Gaussian noise with variances sigma_l, uniform permutation priors, and moduli m_l = Gamma M_l. These are model inputs, not fitted constants, and the algorithms output cluster assignments and reconstructed residues that are then compared against simulated ground truth and against deterministic baselines. No parameter is fitted to a subset of the data and then renamed as a prediction, so the fitted-input pattern does not apply. The self-citations (for example [25] for the deterministic RCRT baseline and [27] for Theorem 3 in the error-correction section) are prior published results with their own stated assumptions; they do not presuppose the present paper's statistical claim, and the paper does not invoke a self-cited uniqueness theorem to force its choice. The questionable step that Assumption 1 is independent of the permutation K, and the wrapped/unwrapped likelihood identity in Lemma 3, are mathematical correctness concerns rather than circular reductions: even if those arguments fail, the output is not equal to the input by construction. Because the derivation is self-contained against external benchmarks and no load-bearing step reduces to a self-citation or to a fitted value, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation does not fit any constants to data; all parameters (Gamma, M_l, sigma_l) are inputs of the generative model. The weakest items are the two mathematical assumptions above, both unproved or false in the regime where the method claims advantage.

assumptions (5)

domain assumption Moduli are of the form m_l = Gamma * M_l with pairwise coprime M_l, so each number has a common residue mu_i modulo Gamma.
This is the standard RCRT structure used throughout (Section II); without it the projection to the small circle in Fig. 1 is undefined.
domain assumption Y_i are i.i.d. uniform on [0, D) with D = Gamma * prod(M_l), and noise Delta_il are i.i.d. Gaussian with variances sigma_l^2.
The Bayesian likelihood and MAP objective in (5)-(6) are built on these priors; the claim of maximal dynamic range relies on D being the lcm.
domain assumption Assumption 1: there exists a cutting point tau on the circle modulo Gamma not inside any interval I_i, with |I_i| < Gamma/2 for every i.
This noise-bound assumption lets the circle be cut and straightened; it is introduced in Section III and conditions the entire MAP derivation.
ad hoc to paper The wrapped-likelihood integral in Eq. (6) equals the unwrapped real-line Gaussian integral in Eq. (13).
This equality is asserted in Lemma 3 without a proof of validity; it is exact only for small noise, not for the heavy-noise regime the paper targets.
ad hoc to paper The event A1 (Assumption 1 holds) is independent of the permutation K, as claimed around Eqs. (11)-(12).
The proof that the integral p(A1|K,R)p(R|K)dR is constant across K rests on the non-sequitur that p(R|K) is a normal density; incorrect clusterings make A1 far less likely, so the assertion is false.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Statistical Robust Chinese Remainder Theorem for Multiple Numbers." pith.science (2026). https://pith.science/paper/R3MA5ACA

@misc{pith2026190900225,
  author       = {Pith},
  title        = {Pith review of: Statistical Robust Chinese Remainder Theorem for Multiple Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R3MA5ACA}},
  note         = {Machine review of arXiv:1909.00225}
}

read the original abstract

Generalized Chinese Remainder Theorem (CRT) is a well-known approach to solve ambiguity resolution related problems. In this paper, we study the robust CRT reconstruction for multiple numbers from a view of statistics. To the best of our knowledge, it is the first rigorous analysis on the underlying statistical model of CRT-based multiple parameter estimation. To address the problem, two novel approaches are established. One is to directly calculate a conditional maximum a posteriori probability (MAP) estimation of the residue clustering, and the other is based on a generalized wrapped Gaussian mixture model to iteratively search for MAP of both estimands and clustering. Residue error correcting codes are introduced to improve the robustness further. Experimental results show that the statistical schemes achieve much stronger robustness compared to state-of-the-art deterministic schemes, especially in heavy-noise scenarios.

Figures

Figures reproduced from arXiv: 1909.00225 by the authors.

**Figure 2.** Illustration for the noise interval [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Illustration for the Step one of Algorithm 2 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Comparison for Success Rate of Robust Reconstruction between the proposed two statistical RCRT and Deterministic [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 7.** Figure 7: Proposed Statistical RCRT-2 with Error Correction [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 6.** Figure 6: The Number of Average Iterations of Algorithm 2 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[25]

Robustness in chinese remainder theorem for multiple numbers and remainder coding

Hanshen Xiao, Yufeng Huang, Yu Ye, and Guoqiang Xiao. Robustness in chinese remainder theorem for multiple numbers and remainder coding. IEEE Transactions on Signal Processing , 66(16):4347–4361, 2018

work page 2018
[1]

Chinese remaindering with errors

Oded Goldreich, Dana Ron, and Madhu Sudan. Chinese remaindering with errors. In Proceedings of the thirty-ﬁrst annual ACM symposium on Theory of computing , pages 225–234. ACM, 1999

work page 1999
[2]

Inequalities

Godfrey Harold Hardy, John Edensor Littlewood, and George Pólya. Inequalities. Cambridge university press, 1988

work page 1988
[3]

Bistatic linear antenna array sar for moving target detection, location, and imaging with two passive airborne radars

Gang Li, Jia Xu, Ying-Ning Peng, and Xiang-Gen Xia. Bistatic linear antenna array sar for moving target detection, location, and imaging with two passive airborne radars. IEEE Transactions on Geoscience and Remote Sensing , 45(3):554–565, 2007

work page 2007
[4]

Location and imaging of moving targets using nonuniform linear antenna array sar

Gang Li, Jia Xu, Ying-Ning Peng, and Xiang-Gen Xia. Location and imaging of moving targets using nonuniform linear antenna array sar. IEEE Transactions on Aerospace and Electronic Systems , 43(3), 2007

work page 2007
[5]

Optimal estimates of two common remainders for a robust generalized chinese remainder theorem

Xiaoping Li, Tingzhu Huang, Qunying Liao, and Xiang-Gen Xia. Optimal estimates of two common remainders for a robust generalized chinese remainder theorem. IEEE Transactions on Signal Processing , 67(7):1824–1837, 2019

work page 2019
[6]

Phase- detection-based range estimation with robust chinese remainder theo- rem

Xiaoping Li, Wenjie Wang, Weile Zhang, and Yunhe Cao. Phase- detection-based range estimation with robust chinese remainder theo- rem. IEEE Transactions on Vehicular Technology, 65(12):10132–10137, 2016

work page 2016
[7]

A robust generalized chinese remainder theorem for two integers

Xiaoping Li, Xiang-Gen Xia, Wenjie Wang, and Wei Wang. A robust generalized chinese remainder theorem for two integers. IEEE Trans- actions on Information Theory , 62(12):7491–7504, 2016

work page 2016

Show all 31 references

[8]

A robust chinese remainder theorem with its applications in frequency estimation from undersampled waveforms

Xiaowei Li, Hong Liang, and Xiang-Gen Xia. A robust chinese remainder theorem with its applications in frequency estimation from undersampled waveforms. IEEE Transactions on Signal Processing , 57(11):4314–4322, 2009

2009
[9]

A fast robust chinese remainder theorem based phase unwrapping algorithm

Xiaowei Li and Xiang-Gen Xia. A fast robust chinese remainder theorem based phase unwrapping algorithm. IEEE Signal Processing Letters , 15:665–668, 2008

2008
[10]

A sharpened dynamic range of a generalized chinese remainder theorem for multiple integers

Huiyong Liao and Xiang-Gen Xia. A sharpened dynamic range of a generalized chinese remainder theorem for multiple integers. IEEE transactions on information theory , 53(1):428–433, 2006

2006
[11]

Nested arrays: A novel approach to array processing with enhanced degrees of freedom

Piya Pal and Palghat P Vaidyanathan. Nested arrays: A novel approach to array processing with enhanced degrees of freedom. IEEE Transactions on Signal Processing , 58(8):4167–4181, 2010

2010
[12]

Multiple-burst error correction with the chinese remainder theorem

Jeremy J Stone. Multiple-burst error correction with the chinese remainder theorem. Journal of the Society for Industrial and Applied Mathematics, 11(1):74–81, 1963

1963
[13]

Direction of arrival estimation using co-prime arrays: A super resolution viewpoint

Zhao Tan, Yonina C Eldar, and Arye Nehorai. Direction of arrival estimation using co-prime arrays: A super resolution viewpoint. IEEE Transactions on Signal Processing , 62(21):5565–5576, 2014

2014
[14]

Sparse sensing with co-prime sam- plers and arrays

Palghat P Vaidyanathan and Piya Pal. Sparse sensing with co-prime sam- plers and arrays. IEEE Transactions on Signal Processing , 59(2):573– 586, 2010

2010
[15]

Robust chinese remainder theorem ranging method based on dual-frequency measurements

Chen Wang, Qinye Yin, and Hongyang Chen. Robust chinese remainder theorem ranging method based on dual-frequency measurements. IEEE Transactions on Vehicular Technology, 60(8):4094–4099, 2011. IEEE TRANSACTIONS ON SIGNAL PROCESSING 12

2011
[16]

Detec- tion, location, and imaging of fast moving targets using multifrequency antenna array sar

Genyuan Wang, Xiang-Gen Xia, Victor C Chen, and RL Fielder. Detec- tion, location, and imaging of fast moving targets using multifrequency antenna array sar. IEEE Transactions on Aerospace and Electronic Systems, 40(1):345–355, 2004

2004
[17]

The largest dynamic range of a generalized chinese remainder theorem for two integers

Wei Wang, Xiaoping Li, Xiang-Gen Xia, and Wenjie Wang. The largest dynamic range of a generalized chinese remainder theorem for two integers. IEEE Signal Processing Letters , 22(2):254–258, 2015

2015
[18]

Maximum likelihood estimation based robust chinese remainder theorem for real numbers and its fast algorithm

Wenjie Wang, Xiaoping Li, Wei Wang, and Xiang-Gen Xia. Maximum likelihood estimation based robust chinese remainder theorem for real numbers and its fast algorithm. IEEE Transactions on Signal Processing, 63(13):3317–3331, 2015

2015
[19]

A closed-form robust chinese remainder theorem and its performance analysis

Wenjie Wang and Xiang-Gen Xia. A closed-form robust chinese remainder theorem and its performance analysis. IEEE Transactions on Signal Processing , 58(11):5655–5666, 2010

2010
[20]

An efﬁcient frequency-determination algorithm from multiple undersampled waveforms

X-G Xia. An efﬁcient frequency-determination algorithm from multiple undersampled waveforms. IEEE Signal Processing Letters , 7(2):34–37, 2000

2000
[21]

On estimation of multiple frequencies in undersampled complex valued waveforms

Xiang-Gen Xia. On estimation of multiple frequencies in undersampled complex valued waveforms. IEEE Transactions on Signal Processing , 47(12):3417–3419, 1999

1999
[22]

A generalized chinese remainder theorem for residue sets with errors and its application in frequency determination from multiple sensors with low sampling rates

Xiang-Gen Xia and Kejing Liu. A generalized chinese remainder theorem for residue sets with errors and its application in frequency determination from multiple sensors with low sampling rates. IEEE Signal Processing Letters , 12(11):768–771, 2005

2005
[23]

Phase unwrapping and a robust chinese remainder theorem

Xiang-Gen Xia and Genyuan Wang. Phase unwrapping and a robust chinese remainder theorem. IEEE Signal Processing Letters, 14(4):247– 250, 2007

2007
[24]

Symmetric polynomial & crt based algorithms for multiple frequency determination from undersampled waveforms

Hanshen Xiao, Cas Cremers, and Hari Krishna Garg. Symmetric polynomial & crt based algorithms for multiple frequency determination from undersampled waveforms. In Signal and Information Processing (GlobalSIP), 2016 IEEE Global Conference on , pages 202–206. IEEE, 2016

2016
[26]

Notes on crt-based robust frequency estimation

Hanshen Xiao and Guoqiang Xiao. Notes on crt-based robust frequency estimation. Signal Processing, 133:13–17, 2017

2017
[27]

On solving ambiguity resolution with robust chinese remainder theorem for multiple numbers

Hanshen Xiao and Guoqiang Xiao. On solving ambiguity resolution with robust chinese remainder theorem for multiple numbers. IEEE Transactions on Vehicular Technology, 68(5):5179–5184, 2019

2019
[28]

Towards robustness in residue number systems

Li Xiao, Xiang-Gen Xia, and Haiye Huo. Towards robustness in residue number systems. IEEE Transactions on Signal Processing , 65(6):1497– 1510, 2017

2017
[29]

Multi-stage robust chi- nese remainder theorem

Li Xiao, Xiang-Gen Xia, and Wenjie Wang. Multi-stage robust chi- nese remainder theorem. IEEE Transactions on Signal Processing , 62(18):4772–4785, 2014

2014
[30]

On solving a generalized chinese remainder theorem in the presence of remainder errors

Guangwu Xu. On solving a generalized chinese remainder theorem in the presence of remainder errors. arXiv preprint arXiv:1409.0121, 2014

2014 arXiv
[31]

Radial velocity retrieval for multichannel sar moving targets with time–space doppler deambiguity

Jia Xu, Zu-Zhen Huang, Zhi-Rui Wang, Li Xiao, Xiang-Gen Xia, and Teng Long. Radial velocity retrieval for multichannel sar moving targets with time–space doppler deambiguity. IEEE Transactions on Geoscience and Remote Sensing , 56(1):35–48, 2018

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [25]

Robustness in chinese remainder theorem for multiple numbers and remainder coding

Hanshen Xiao, Yufeng Huang, Yu Ye, and Guoqiang Xiao. Robustness in chinese remainder theorem for multiple numbers and remainder coding. IEEE Transactions on Signal Processing , 66(16):4347–4361, 2018

work page 2018

[2] [1]

Chinese remaindering with errors

Oded Goldreich, Dana Ron, and Madhu Sudan. Chinese remaindering with errors. In Proceedings of the thirty-ﬁrst annual ACM symposium on Theory of computing , pages 225–234. ACM, 1999

work page 1999

[3] [2]

Inequalities

Godfrey Harold Hardy, John Edensor Littlewood, and George Pólya. Inequalities. Cambridge university press, 1988

work page 1988

[4] [3]

Bistatic linear antenna array sar for moving target detection, location, and imaging with two passive airborne radars

Gang Li, Jia Xu, Ying-Ning Peng, and Xiang-Gen Xia. Bistatic linear antenna array sar for moving target detection, location, and imaging with two passive airborne radars. IEEE Transactions on Geoscience and Remote Sensing , 45(3):554–565, 2007

work page 2007

[5] [4]

Location and imaging of moving targets using nonuniform linear antenna array sar

Gang Li, Jia Xu, Ying-Ning Peng, and Xiang-Gen Xia. Location and imaging of moving targets using nonuniform linear antenna array sar. IEEE Transactions on Aerospace and Electronic Systems , 43(3), 2007

work page 2007

[6] [5]

Optimal estimates of two common remainders for a robust generalized chinese remainder theorem

Xiaoping Li, Tingzhu Huang, Qunying Liao, and Xiang-Gen Xia. Optimal estimates of two common remainders for a robust generalized chinese remainder theorem. IEEE Transactions on Signal Processing , 67(7):1824–1837, 2019

work page 2019

[7] [6]

Phase- detection-based range estimation with robust chinese remainder theo- rem

Xiaoping Li, Wenjie Wang, Weile Zhang, and Yunhe Cao. Phase- detection-based range estimation with robust chinese remainder theo- rem. IEEE Transactions on Vehicular Technology, 65(12):10132–10137, 2016

work page 2016

[8] [7]

A robust generalized chinese remainder theorem for two integers

Xiaoping Li, Xiang-Gen Xia, Wenjie Wang, and Wei Wang. A robust generalized chinese remainder theorem for two integers. IEEE Trans- actions on Information Theory , 62(12):7491–7504, 2016

work page 2016