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Robust Multidimensional Chinese Remainder Theorem (MD-CRT) with Non-Diagonal Moduli and Multi-Stage Framework

Guangpu Guo, Xiang-Gen Xia

Abstract

The Chinese remainder theorem (CRT) provides an efficient way to reconstruct an integer from its remainders modulo several integer moduli, and has been widely applied in signal processing and information theory. Its multidimensional extension (MD-CRT) generalizes this principle to integer vectors and integer matrix moduli, enabling reconstruction in multidimensional signal processing scenarios. However, since matrices are generally non-commutative, the multidimensional extension introduces new theoretical and algorithmic challenges. When all matrix moduli are diagonal, the system is equivalent to applying the one-dimensional CRT independently along each dimension. This work first investigates whether non-diagonal (non-separable) moduli offer fundamental advantages over traditional diagonal ones. We show that under the same determinant constraint, non-diagonal matrices do not increase the dynamic range but yield more balanced and better-conditioned sampling patterns. More importantly, they generate lattices with longer shortest vectors, leading to higher robustness to vector remainder errors, compared to diagonal ones. To further improve the robustness, we develop a multi-stage robust MD-CRT framework that improves the robustness level without reducing the dynamic range. Due to the multidimensional nature and modulo matrix forms, it is challenging and not straightforward to extend the existing one-dimensional multi-stage robust CRT. In this paper, we obtain a new condition for matrix moduli, which can be easily checked, such that a multi-stage robust MD-CRT can be implemented. Both theoretical analysis and simulation results demonstrate that the proposed multi-stage robust MD-CRT achieves stronger error tolerance and more reliable reconstruction under erroneous vector remainders than that of single-stage robust MD-CRT.

Robust Multidimensional Chinese Remainder Theorem (MD-CRT) with Non-Diagonal Moduli and Multi-Stage Framework

Abstract

The Chinese remainder theorem (CRT) provides an efficient way to reconstruct an integer from its remainders modulo several integer moduli, and has been widely applied in signal processing and information theory. Its multidimensional extension (MD-CRT) generalizes this principle to integer vectors and integer matrix moduli, enabling reconstruction in multidimensional signal processing scenarios. However, since matrices are generally non-commutative, the multidimensional extension introduces new theoretical and algorithmic challenges. When all matrix moduli are diagonal, the system is equivalent to applying the one-dimensional CRT independently along each dimension. This work first investigates whether non-diagonal (non-separable) moduli offer fundamental advantages over traditional diagonal ones. We show that under the same determinant constraint, non-diagonal matrices do not increase the dynamic range but yield more balanced and better-conditioned sampling patterns. More importantly, they generate lattices with longer shortest vectors, leading to higher robustness to vector remainder errors, compared to diagonal ones. To further improve the robustness, we develop a multi-stage robust MD-CRT framework that improves the robustness level without reducing the dynamic range. Due to the multidimensional nature and modulo matrix forms, it is challenging and not straightforward to extend the existing one-dimensional multi-stage robust CRT. In this paper, we obtain a new condition for matrix moduli, which can be easily checked, such that a multi-stage robust MD-CRT can be implemented. Both theoretical analysis and simulation results demonstrate that the proposed multi-stage robust MD-CRT achieves stronger error tolerance and more reliable reconstruction under erroneous vector remainders than that of single-stage robust MD-CRT.

Paper Structure

This paper contains 15 sections, 11 theorems, 71 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamic Range Analysis of MD-CRT
  4. Multidimensional Undersampling and MD-CRT
  5. Dynamic Range Analysis
  6. Sampling Advantage of Non-Diagonal Matrix Moduli
  7. Single-Stage Robust MD-CRT
  8. Single-Stage Robust MD-CRT
  9. Robustness Advantage of Non-Diagonal Matrix Moduli
  10. Multi-Stage Robust MD-CRT
  11. Motivating Example
  12. A Sufficient Condition for Multi-Stage Robustness
  13. A Multi-Stage Robust Reconstruction Framework
  14. Simulations
  15. Conclusion

Key Result

Proposition 1

Let $\mathbf{M}_1,\mathbf{M}_2,\ldots,\mathbf{M}_L$ be arbitrary nonsingular integer matrices of size $D\times D$, and let $\mathbf{R}$ be any lcrm of them. For any integer vector $\mathbf{f}\in\mathbb{Z}^{D}$, it can be uniquely determined from its vector remainders $\mathbf{r}_i\equiv\mathbf{f}\bm

Figures (3)

  • Figure 1: The robustly determinable range in Example \ref{['ex:shiftedFPD']}
  • Figure 2: Mean reconstruction error versus vector remainder error bound $\tau$ for diagonal, non-diagonal, and two-stage systems.
  • Figure 3: Mean reconstruction error versus vector remainder error bound $\tau$ for the single-stage and two-stage systems using the six matrix moduli in Section \ref{['s5']}.

Theorems & Definitions (20)

  • Proposition 1: MD-CRT MD1
  • Lemma 1
  • proof
  • Lemma 2: Lemma 1 in guo25
  • Lemma 3: Lemma 2 in MD1
  • Theorem 1
  • proof
  • Corollary 1
  • Proposition 2: Robust MD-CRT from MD2
  • Example 1
  • ...and 10 more