将完美重构滤波器组分解为因果提升矩阵：一种丢番图方法
Factoring Perfect Reconstruction Filter Banks into Causal Lifting Matrices: A Diophantine Approach
AI总结：
本文基于多项式环上二元线性丢番图方程理论构建因果双通道FIR完美重构传递矩阵的因果提升分解，提出因果互补算法，为Daubechies-Sweldens非因果方案提供因果替代且更通用。
摘要：
本文利用多项式环上二元线性丢番图方程的基础理论，为因果双通道FIR完美重构传递矩阵和小波变换构建因果提升分解（初等矩阵分解）。丢番图方法生成满足特定多项式降次不等式的因果分解，进而提出名为因果互补算法的新分解策略。该策略为Daubechies和Sweldens使用Laurent多项式扩展欧几里得算法开发的非因果提升方案提供了因果（即多项式，可实现）替代方案。新方法用高斯消元法替代欧几里得算法，高斯消元法采用多项式除法的轻微泛化，确保满足用户指定整除约束的商的存在性和唯一性。因果互补算法比欧几里得算法方法的因果版本更通用，能生成多项式欧几里得算法无法得到的额外因果提升分解。
英文摘要：
The elementary theory of bivariate linear Diophantine equations over polynomial rings is used to construct causal lifting factorizations (elementary matrix decompositions) for causal two-channel FIR perfect reconstruction transfer matrices and wavelet transforms. The Diophantine approach generates causal factorizations satisfying certain polynomial degree-reducing inequalities, enabling a new factorization strategy called the Causal Complementation Algorithm. This provides a causal (i.e., polynomial, hence realizable) alternative to the noncausal lifting scheme developed by Daubechies and Sweldens using the Extended Euclidean Algorithm for Laurent polynomials. The new approach replaces the Euclidean Algorithm with Gaussian elimination employing a slight generalization of polynomial division that ensures existence and uniqueness of quotients whose remainders satisfy user-specified divisibility constraints. The Causal Complementation Algorithm is shown to be more general than the causal version of the Euclidean Algorithm approach by generating additional causal lifting factorizations beyond those obtainable using the polynomial Euclidean Algorithm.
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